id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-000401 | Calculus: Limits — Difference Quotients | 1 | Explain why your operations are valid: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 342}\frac{x^2-(342)^2}{x-(342)}.$$
(a) Evaluate the limit by algebraic simplification.... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{684}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=684$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 684$. (Here the result is $\boxed{684}$.) |
math-000402 | Calculus: Limits — Removable Discontinuities | 1 | Exercise: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -360}\frac{x^2-(-360)^2}{x-(-360)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-720}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-720$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -720$. (Here the result is $\boxed{-720}$.) |
math-000403 | Calculus: Limits — Algebraic Simplification | 1 | Work carefully and justify each inference: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 328}\frac{x^2-(328)^2}{x-(328)}.$$
(a) Evaluate the limit by algebraic simpli... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{656}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=656$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 656$. |
math-000404 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Warm-up: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -12}\frac{x^2-(-12)^2}{x-(-12)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-24}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-24$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustne... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -24$. (Here the result is $\boxed{-24}$.) |
math-000405 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Proceed methodically: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 130}\frac{x^2-(130)^2}{x-(130)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and comput... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{260}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=260$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 260$. |
math-000406 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Solve and justify each step: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -189}\frac{x^2-(-189)^2}{x-(-189)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-378}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-378$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"rob... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -378$. (Here the result is $\boxed{-378}$.) |
math-000407 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Be explicit about assumptions: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 37}\frac{x^2-(37)^2}{x-(37)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinte... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(37)^2=(x-(37))(x+(37))$.",
"Step 2: For $x\\neq 37$, cancel to get $\\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{74}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=74$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem w... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 74$. |
math-000408 | Calculus: Limits — Removable Discontinuities | 1 | Solve and justify each step: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -313}\frac{x^2-(-313)^2}{x-(-313)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-626}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-626$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -626$. (Here the result is $\boxed{-626}$.) |
math-000409 | Calculus: Limits — Removable Discontinuities | 1 | Start by stating any domain restrictions: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 280}\frac{x^2-(280)^2}{x-(280)}.$$
(a) Evaluate the limit by algebraic simplif... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{560}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=560$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 560$. (Here the result is $\boxed{560}$.) |
math-000410 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Use two approaches if possible: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -257}\frac{x^2-(-257)^2}{x-(-257)}.$$
(a) Evaluate the limit by algebraic simplification... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-514}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-514$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -514$. |
math-000411 | Calculus: Limits — Algebraic Simplification | 1 | Provide both a computational and a conceptual explanation: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 218}\frac{x^2-(218)^2}{x-(218)}.$$
(a) Evaluate the limit by algebraic simplifi... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{436}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=436$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the p... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 436$. (Here the result is $\boxed{436}$.) |
math-000412 | Calculus: Limits — Removable Discontinuities | 1 | Task: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 85}\frac{x^2-(85)^2}{x-(85)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and co... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(85)^2=(x-(85))(x+(85))$.",
"Step 2: For $x\\neq 85$, cancel to get $\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{170}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=170$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 170$. |
math-000413 | Calculus: Limits — Algebraic Simplification | 1 | Keep the final answer in boxed form: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 302}\frac{x^2-(302)^2}{x-(302)}.$$
(a) Evaluate the limit by algebraic simplification.
(... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{604}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=604$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 604$. (Here the result is $\boxed{604}$.) |
math-000414 | Calculus: Limits — Removable Discontinuities | 1 | Solve and then verify: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -294}\frac{x^2-(-294)^2}{x-(-294)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterp... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-588}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-588$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensit... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -588$. (Here the result is $\boxed{-588}$.) |
math-000415 | Calculus: Limits — Algebraic Simplification | 1 | Problem: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -89}\frac{x^2-(-89)^2}{x-(-89)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as ... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-89)^2=(x-(-89))(x+(-89))$.",
"Step 2: For $x\\neq -89$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-178}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-178$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -178$. (Here the result is $\boxed{-178}$.) |
math-000416 | Calculus: Limits — Algebraic Simplification | 1 | Complete the analysis: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -315}\frac{x^2-(-315)^2}{x-(-315)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and co... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-315)^2=(x-(-315))(x+(-315))$.",
"Step 2: For $x\\neq -315$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-630}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-630$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -630$. (Here the result is $\boxed{-630}$.) |
math-000417 | Calculus: Limits — Removable Discontinuities | 1 | Solve and include a self-check: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -375}\frac{x^2-(-375)^2}{x-(-375)}.$$
(a) Evaluate the limit by algebraic simplification... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-750}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-750$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -750$. (Here the result is $\boxed{-750}$.) |
math-000418 | Calculus: Limits — Removable Discontinuities | 1 | Proceed methodically: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 52}\frac{x^2-(52)^2}{x-(52)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a d... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(52)^2=(x-(52))(x+(52))$.",
"Step 2: For $x\\neq 52$, cancel to get $\\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{104}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=104$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the p... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 104$. |
math-000419 | Calculus: Limits — Algebraic Simplification | 1 | Solve and then verify: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -165}\frac{x^2-(-165)^2}{x-(-165)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterp... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-330}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-330$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -330$. (Here the result is $\boxed{-330}$.) |
math-000420 | Calculus: Limits — Removable Discontinuities | 1 | Solve and sanity-check: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 226}\frac{x^2-(226)^2}{x-(226)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reint... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{452}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=452$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 452$. (Here the result is $\boxed{452}$.) |
math-000421 | Calculus: Limits — Difference Quotients | 1 | Task: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 231}\frac{x^2-(231)^2}{x-(231)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that way.
(... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(231)^2=(x-(231))(x+(231))$.",
"Step 2: For $x\\neq 231$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{462}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=462$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 462$. (Here the result is $\boxed{462}$.) |
math-000422 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Derive the result step-by-step: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 186}\frac{x^2-(186)^2}{x-(186)}.$$
(a) Evaluate the limit by algebraic simplification.
(... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{372}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=372$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 372$. (Here the result is $\boxed{372}$.) |
math-000423 | Calculus: Limits — Difference Quotients | 1 | Track units/moduli carefully: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -146}\frac{x^2-(-146)^2}{x-(-146)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-292}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-292$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robust... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -292$. (Here the result is $\boxed{-292}$.) |
math-000424 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Complete the analysis: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -16}\frac{x^2-(-16)^2}{x-(-16)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-32}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-32$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -32$. |
math-000425 | Calculus: Limits — Algebraic Simplification | 1 | Work carefully and justify each inference: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 136}\frac{x^2-(136)^2}{x-(136)}.$$
(a) Evaluate the limit by algebraic simplificat... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{272}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=272$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 272$. |
math-000426 | Calculus: Limits — Removable Discontinuities | 1 | Provide both a computational and a conceptual explanation: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -284}\frac{x^2-(-284)^2}{x-(-284)}.$$
(a) Evaluate the limit by al... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-568}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-568$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Genera... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -568$. |
math-000427 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Explain why your operations are valid: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -182}\frac{x^2-(-182)^2}{x-(-182)}.$$
(a) Evaluate the limit by algebraic simplificati... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-364}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-364$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Genera... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -364$. |
math-000428 | Calculus: Limits — Removable Discontinuities | 1 | Be explicit about assumptions: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -169}\frac{x^2-(-169)^2}{x-(-169)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivativ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-338}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-338$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -338$. (Here the result is $\boxed{-338}$.) |
math-000429 | Calculus: Limits — Algebraic Simplification | 1 | Be explicit about assumptions: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 360}\frac{x^2-(360)^2}{x-(360)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{720}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=720$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitiv... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 720$. (Here the result is $\boxed{720}$.) |
math-000430 | Calculus: Limits — Difference Quotients | 1 | Proceed methodically: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 397}\frac{x^2-(397)^2}{x-(397)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and comput... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{794}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=794$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 794$. |
math-000431 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Answer using clear logical steps: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 349}\frac{x^2-(349)^2}{x-(349)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{698}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=698$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 698$. |
math-000432 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Give reasoning, not just computation: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -1}\frac{x^2-(-1)^2}{x-(-1)}.$$
(a) Evaluate the limit by algebraic simplification.
(b)... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-2}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-2$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitivity anal... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -2$. (Here the result is $\boxed{-2}$.) |
math-000433 | Calculus: Limits — Difference Quotients | 1 | Track quantifiers carefully: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 172}\frac{x^2-(172)^2}{x-(172)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reint... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{344}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=344$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 344$. (Here the result is $\boxed{344}$.) |
math-000434 | Calculus: Limits — Algebraic Simplification | 1 | Be explicit about assumptions: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -343}\frac{x^2-(-343)^2}{x-(-343)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivativ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-686}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-686$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -686$. (Here the result is $\boxed{-686}$.) |
math-000435 | Calculus: Limits — Algebraic Simplification | 1 | Find the exact value: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -167}\frac{x^2-(-167)^2}{x-(-167)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpr... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-167)^2=(x-(-167))(x+(-167))$.",
"Step 2: For $x\\neq -167$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-334}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-334$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -334$. (Here the result is $\boxed{-334}$.) |
math-000436 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Start by stating any domain restrictions: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -148}\frac{x^2-(-148)^2}{x-(-148)}.$$
(a) Evaluate the limit by algebraic simplific... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-296}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-296$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"rob... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -296$. |
math-000437 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Checkpoint: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -347}\frac{x^2-(-347)^2}{x-(-347)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it th... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-694}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-694$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -694$. |
math-000438 | Calculus: Limits — Difference Quotients | 1 | Give a theorem-based solution: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 290}\frac{x^2-(290)^2}{x-(290)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{580}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=580$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 580$. |
math-000439 | Calculus: Limits — Difference Quotients | 1 | Exercise: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 340}\frac{x^2-(340)^2}{x-(340)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the lim... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{680}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=680$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitiv... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 680$. (Here the result is $\boxed{680}$.) |
math-000440 | Calculus: Limits — Algebraic Simplification | 1 | Compute the requested quantity: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -395}\frac{x^2-(-395)^2}{x-(-395)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-790}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-790$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -790$. (Here the result is $\boxed{-790}$.) |
math-000441 | Calculus: Limits — Difference Quotients | 1 | Answer using clear logical steps: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -70}\frac{x^2-(-70)^2}{x-(-70)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-70)^2=(x-(-70))(x+(-70))$.",
"Step 2: For $x\\neq -70$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-140}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-140$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -140$. |
math-000442 | Calculus: Limits — Algebraic Simplification | 1 | Solve and sanity-check: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 372}\frac{x^2-(372)^2}{x-(372)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and comp... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{744}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=744$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 744$. (Here the result is $\boxed{744}$.) |
math-000443 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Write the solution set clearly: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -334}\frac{x^2-(-334)^2}{x-(-334)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivati... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-668}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-668$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensit... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -668$. |
math-000444 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Track units/moduli carefully: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 128}\frac{x^2-(128)^2}{x-(128)}.$$
(a) Evaluate the limit by algebraic simplification.
(b)... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(128)^2=(x-(128))(x+(128))$.",
"Step 2: For $x\\neq 128$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{256}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=256$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 256$. (Here the result is $\boxed{256}$.) |
math-000445 | Calculus: Limits — Algebraic Simplification | 1 | Give reasoning, not just computation: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -181}\frac{x^2-(-181)^2}{x-(-181)}.$$
(a) Evaluate the limit by algebraic simplificatio... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-181)^2=(x-(-181))(x+(-181))$.",
"Step 2: For $x\\neq -181$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-362}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-362$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"rob... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -362$. (Here the result is $\boxed{-362}$.) |
math-000446 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Derive the result step-by-step: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 177}\frac{x^2-(177)^2}{x-(177)}.$$
(a) Evaluate the limit by algebraic simplification.
(... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{354}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=354$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitivity an... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 354$. (Here the result is $\boxed{354}$.) |
math-000447 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Solve and then verify: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -338}\frac{x^2-(-338)^2}{x-(-338)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterp... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-676}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-676$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -676$. |
math-000448 | Calculus: Limits — Removable Discontinuities | 1 | Try to avoid pattern-matching; explain why: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 366}\frac{x^2-(366)^2}{x-(366)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{732}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=732$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 732$. |
math-000449 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Proceed methodically: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -106}\frac{x^2-(-106)^2}{x-(-106)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and com... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-212}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-212$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"rob... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -212$. (Here the result is $\boxed{-212}$.) |
math-000450 | Calculus: Limits — Algebraic Simplification | 1 | Solve and justify each step: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 270}\frac{x^2-(270)^2}{x-(270)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the li... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{540}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=540$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 540$. (Here the result is $\boxed{540}$.) |
math-000451 | Calculus: Limits — Removable Discontinuities | 1 | Provide a rigorous solution: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -316}\frac{x^2-(-316)^2}{x-(-316)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-632}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-632$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -632$. (Here the result is $\boxed{-632}$.) |
math-000452 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Task: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -144}\frac{x^2-(-144)^2}{x-(-144)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that way... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-144)^2=(x-(-144))(x+(-144))$.",
"Step 2: For $x\\neq -144$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-288}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-288$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"rob... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -288$. |
math-000453 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Where appropriate, name the theorem you use: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 314}\frac{x^2-(314)^2}{x-(314)}.$$
(a) Evaluate the limit by algebraic simp... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{628}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=628$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitivity an... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 628$. (Here the result is $\boxed{628}$.) |
math-000454 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Solve and justify each step: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -20}\frac{x^2-(-20)^2}{x-(-20)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the li... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-40}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-40$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -40$. |
math-000455 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Proceed methodically: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -256}\frac{x^2-(-256)^2}{x-(-256)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpr... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-512}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-512$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -512$. |
math-000456 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Derive the result step-by-step: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 228}\frac{x^2-(228)^2}{x-(228)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Re... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(228)^2=(x-(228))(x+(228))$.",
"Step 2: For $x\\neq 228$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{456}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=456$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 456$. (Here the result is $\boxed{456}$.) |
math-000457 | Calculus: Limits — Removable Discontinuities | 1 | Exercise: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -325}\frac{x^2-(-325)^2}{x-(-325)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-650}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-650$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robust... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -650$. (Here the result is $\boxed{-650}$.) |
math-000458 | Calculus: Limits — Difference Quotients | 1 | Work this out carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 63}\frac{x^2-(63)^2}{x-(63)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{126}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=126$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 126$. (Here the result is $\boxed{126}$.) |
math-000459 | Calculus: Limits — Removable Discontinuities | 1 | Give reasoning, not just computation: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 10}\frac{x^2-(10)^2}{x-(10)}.$$
(a) Evaluate the limit by algebraic simplification.
(b)... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{20}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=20$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustn... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 20$. (Here the result is $\boxed{20}$.) |
math-000460 | Calculus: Limits — Algebraic Simplification | 1 | Determine the requested value: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 350}\frac{x^2-(350)^2}{x-(350)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(350)^2=(x-(350))(x+(350))$.",
"Step 2: For $x\\neq 350$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{700}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=700$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 700$. (Here the result is $\boxed{700}$.) |
math-000461 | Calculus: Limits — Removable Discontinuities | 1 | Problem: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -77}\frac{x^2-(-77)^2}{x-(-77)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limi... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-77)^2=(x-(-77))(x+(-77))$.",
"Step 2: For $x\\neq -77$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-154}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-154$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Genera... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -154$. (Here the result is $\boxed{-154}$.) |
math-000462 | Calculus: Limits — Algebraic Simplification | 1 | Give a fully justified solution: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 309}\frac{x^2-(309)^2}{x-(309)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(309)^2=(x-(309))(x+(309))$.",
"Step 2: For $x\\neq 309$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{618}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=618$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 618$. (Here the result is $\boxed{618}$.) |
math-000463 | Calculus: Limits — Removable Discontinuities | 1 | Carefully track domains: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 77}\frac{x^2-(77)^2}{x-(77)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinter... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{154}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=154$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 154$. (Here the result is $\boxed{154}$.) |
math-000464 | Calculus: Limits — Algebraic Simplification | 1 | Show all reasoning: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 116}\frac{x^2-(116)^2}{x-(116)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret th... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(116)^2=(x-(116))(x+(116))$.",
"Step 2: For $x\\neq 116$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{232}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=232$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 232$. (Here the result is $\boxed{232}$.) |
math-000465 | Calculus: Limits — Algebraic Simplification | 1 | Give a fully justified solution: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -60}\frac{x^2-(-60)^2}{x-(-60)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-60)^2=(x-(-60))(x+(-60))$.",
"Step 2: For $x\\neq -60$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-120}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-120$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -120$. (Here the result is $\boxed{-120}$.) |
math-000466 | Calculus: Limits — Algebraic Simplification | 1 | Explain what is being counted/optimized: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 9}\frac{x^2-(9)^2}{x-(9)}.$$
(a) Evaluate the limit by algebraic simplification.
(b)... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{18}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=18$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustn... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 18$. |
math-000467 | Calculus: Limits — Algebraic Simplification | 1 | Explain each transformation: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 217}\frac{x^2-(217)^2}{x-(217)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reint... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{434}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=434$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 434$. (Here the result is $\boxed{434}$.) |
math-000468 | Calculus: Limits — Algebraic Simplification | 1 | Answer using clear logical steps: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 176}\frac{x^2-(176)^2}{x-(176)}.$$
(a) Evaluate the limit by algebraic simplification.... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{352}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=352$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 352$. |
math-000469 | Calculus: Limits — Algebraic Simplification | 1 | Explain each transformation: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -92}\frac{x^2-(-92)^2}{x-(-92)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-184}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-184$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -184$. |
math-000470 | Calculus: Limits — Algebraic Simplification | 1 | Give reasoning, not just computation: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 346}\frac{x^2-(346)^2}{x-(346)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a deriv... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(346)^2=(x-(346))(x+(346))$.",
"Step 2: For $x\\neq 346$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{692}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=692$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 692$. (Here the result is $\boxed{692}$.) |
math-000471 | Calculus: Limits — Difference Quotients | 1 | Task: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -28}\frac{x^2-(-28)^2}{x-(-28)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-56}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-56$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -56$. (Here the result is $\boxed{-56}$.) |
math-000472 | Calculus: Limits — Algebraic Simplification | 1 | Work this out carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 61}\frac{x^2-(61)^2}{x-(61)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as ... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(61)^2=(x-(61))(x+(61))$.",
"Step 2: For $x\\neq 61$, cancel to get $\\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{122}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=122$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 122$. |
math-000473 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Give a fully justified solution: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -398}\frac{x^2-(-398)^2}{x-(-398)}.$$
(a) Evaluate the limit by algebraic simplification.
(b... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-796}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-796$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -796$. (Here the result is $\boxed{-796}$.) |
math-000474 | Calculus: Limits — Algebraic Simplification | 1 | Give a theorem-based solution: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 327}\frac{x^2-(327)^2}{x-(327)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{654}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=654$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 654$. (Here the result is $\boxed{654}$.) |
math-000475 | Calculus: Limits — Removable Discontinuities | 1 | Explain each transformation: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 307}\frac{x^2-(307)^2}{x-(307)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(307)^2=(x-(307))(x+(307))$.",
"Step 2: For $x\\neq 307$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{614}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=614$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitiv... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 614$. |
math-000476 | Calculus: Limits — Algebraic Simplification | 1 | Determine the requested value: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -365}\frac{x^2-(-365)^2}{x-(-365)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-730}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-730$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"rob... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -730$. |
math-000477 | Calculus: Limits — Difference Quotients | 1 | Question: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 69}\frac{x^2-(69)^2}{x-(69)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{138}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=138$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 138$. |
math-000478 | Calculus: Limits — Algebraic Simplification | 1 | Derive the result step-by-step: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 16}\frac{x^2-(16)^2}{x-(16)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{32}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=32$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness note:... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 32$. |
math-000479 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Give an answer and a quick verification: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 308}\frac{x^2-(308)^2}{x-(308)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a de... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(308)^2=(x-(308))(x+(308))$.",
"Step 2: For $x\\neq 308$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{616}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=616$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_ana... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 616$. |
math-000480 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Solve and then verify: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 301}\frac{x^2-(301)^2}{x-(301)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compu... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{602}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=602$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 602$. (Here the result is $\boxed{602}$.) |
math-000481 | Calculus: Limits — Removable Discontinuities | 1 | Keep the final answer in boxed form: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -330}\frac{x^2-(-330)^2}{x-(-330)}.$$
(a) Evaluate the limit by algebraic simplific... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-330)^2=(x-(-330))(x+(-330))$.",
"Step 2: For $x\\neq -330$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-660}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-660$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -660$. (Here the result is $\boxed{-660}$.) |
math-000482 | Calculus: Limits — Algebraic Simplification | 1 | Provide both a computational and a conceptual explanation: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 305}\frac{x^2-(305)^2}{x-(305)}.$$
(a) Evaluate the limit by algeb... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(305)^2=(x-(305))(x+(305))$.",
"Step 2: For $x\\neq 305$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{610}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=610$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 610$. (Here the result is $\boxed{610}$.) |
math-000483 | Calculus: Limits — Difference Quotients | 1 | Prompt: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -381}\frac{x^2-(-381)^2}{x-(-381)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit a... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-762}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-762$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -762$. (Here the result is $\boxed{-762}$.) |
math-000484 | Calculus: Limits — Removable Discontinuities | 1 | Solve and include a self-check: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 36}\frac{x^2-(36)^2}{x-(36)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{72}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=72$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustn... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 72$. (Here the result is $\boxed{72}$.) |
math-000485 | Calculus: Limits — Algebraic Simplification | 1 | Solve with verification: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -250}\frac{x^2-(-250)^2}{x-(-250)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinte... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-500}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-500$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robust... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -500$. |
math-000486 | Calculus: Limits — Removable Discontinuities | 1 | Solve and sanity-check: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -252}\frac{x^2-(-252)^2}{x-(-252)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinter... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-252)^2=(x-(-252))(x+(-252))$.",
"Step 2: For $x\\neq -252$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-504}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-504$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -504$. (Here the result is $\boxed{-504}$.) |
math-000487 | Calculus: Limits — Algebraic Simplification | 1 | Solve (and briefly cross-validate): Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -8}\frac{x^2-(-8)^2}{x-(-8)}.$$
(a) Evaluate the limit by algebraic simplification.
... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-8)^2=(x-(-8))(x+(-8))$.",
"Step 2: For $x\\neq -8$, cancel to get $\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-16}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-16$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -16$. (Here the result is $\boxed{-16}$.) |
math-000488 | Calculus: Limits — Removable Discontinuities | 1 | Give a fully justified solution: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 380}\frac{x^2-(380)^2}{x-(380)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(380)^2=(x-(380))(x+(380))$.",
"Step 2: For $x\\neq 380$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{760}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=760$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 760$. (Here the result is $\boxed{760}$.) |
math-000489 | Calculus: Limits — Difference Quotients | 1 | Show all reasoning: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -177}\frac{x^2-(-177)^2}{x-(-177)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-177)^2=(x-(-177))(x+(-177))$.",
"Step 2: For $x\\neq -177$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-354}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-354$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Genera... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -354$. (Here the result is $\boxed{-354}$.) |
math-000490 | Calculus: Limits — Difference Quotients | 1 | Provide both a computational and a conceptual explanation: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -280}\frac{x^2-(-280)^2}{x-(-280)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterp... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-280)^2=(x-(-280))(x+(-280))$.",
"Step 2: For $x\\neq -280$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-560}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-560$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"rob... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -560$. (Here the result is $\boxed{-560}$.) |
math-000491 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Proceed methodically: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -43}\frac{x^2-(-43)^2}{x-(-43)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and comput... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-43)^2=(x-(-43))(x+(-43))$.",
"Step 2: For $x\\neq -43$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-86}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-86$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitivity an... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -86$. (Here the result is $\boxed{-86}$.) |
math-000492 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Proceed methodically: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -115}\frac{x^2-(-115)^2}{x-(-115)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rein... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-230}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-230$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensit... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -230$. |
math-000493 | Calculus: Limits — Removable Discontinuities | 1 | Answer using clear logical steps: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 183}\frac{x^2-(183)^2}{x-(183)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{366}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=366$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitiv... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 366$. (Here the result is $\boxed{366}$.) |
math-000494 | Calculus: Limits — Difference Quotients | 1 | Keep the final answer in boxed form: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -295}\frac{x^2-(-295)^2}{x-(-295)}.$$
(a) Evaluate the limit by algebraic simplific... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-590}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-590$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -590$. (Here the result is $\boxed{-590}$.) |
math-000495 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Complete the analysis: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 300}\frac{x^2-(300)^2}{x-(300)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinte... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{600}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=600$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 600$. (Here the result is $\boxed{600}$.) |
math-000496 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Determine the requested value: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -312}\frac{x^2-(-312)^2}{x-(-312)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-312)^2=(x-(-312))(x+(-312))$.",
"Step 2: For $x\\neq -312$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-624}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-624$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -624$. |
math-000497 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | State any required conditions first: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -358}\frac{x^2-(-358)^2}{x-(-358)}.$$
(a) Evaluate the limit by algebraic simplification... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-358)^2=(x-(-358))(x+(-358))$.",
"Step 2: For $x\\neq -358$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-716}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-716$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -716$. (Here the result is $\boxed{-716}$.) |
math-000498 | Calculus: Limits — Algebraic Simplification | 1 | Provide a rigorous solution: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 385}\frac{x^2-(385)^2}{x-(385)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(385)^2=(x-(385))(x+(385))$.",
"Step 2: For $x\\neq 385$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{770}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=770$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the p... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 770$. (Here the result is $\boxed{770}$.) |
math-000499 | Calculus: Limits — Algebraic Simplification | 1 | Explain what is being counted/optimized: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -207}\frac{x^2-(-207)^2}{x-(-207)}.$$
(a) Evaluate the limit by algebraic simplifica... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-414}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-414$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -414$. |
math-000500 | Calculus: Limits — Algebraic Simplification | 1 | Make each step logically reversible (or explain if not): Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 51}\frac{x^2-(51)^2}{x-(51)}.$$
(a) Evaluate the limit by algebraic simplificatio... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{102}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=102$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 102$. (Here the result is $\boxed{102}$.) |
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