id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-002401 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to -157}\frac{x^2-(-157)^2}{x-(-157)}.$$
(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: $\\boxed{-314}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-314$. 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... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -314$. |
math-002402 | Calculus: Limits — Removable Discontinuities | 2 | Keep the final answer in boxed form: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -356}\frac{x^2-(-356)^2}{x-(-356)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a der... | [
{
"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: $\\boxed{-712}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-712$. 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 $2a= -712$. |
math-002403 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Work carefully and justify each inference: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -112}\frac{x^2-(-112)^2}{x-(-112)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as... | [
{
"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: $\\boxed{-224}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-224$. 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 $2a= -224$. |
math-002404 | Calculus: Limits — Difference Quotients | 2 | Proceed methodically: 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:
$$\lim_{x\to 281}\frac{x^2-(281)^2}{x-(281)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"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: $\\boxed{562}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=562$. 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 $2a= 562$. (Here the result is $\boxed{562}$.) |
math-002405 | Calculus: Limits — Difference Quotients | 2 | Explain why your operations are valid: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 35}\frac{x^2-(35)^2}{x-(35)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivat... | [
{
"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: $\\boxed{70}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=70$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analy... | [
{
"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 $2a= 70$. |
math-002406 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Solve and sanity-check: 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:
$$\lim_{x\to 199}\frac{x^2-(199)^2}{x-(199)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{398}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=398$. 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 $2a= 398$. |
math-002407 | Calculus: Limits — Difference Quotients | 2 | Solve and then verify: 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:
$$\lim_{x\to 151}\frac{x^2-(151)^2}{x-(151)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as... | [
{
"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: $\\boxed{302}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=302$. 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 $2a= 302$. (Here the result is $\boxed{302}$.) |
math-002408 | Calculus: Limits — Algebraic Simplification | 2 | Find the exact value: 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:
$$\lim_{x\to 25}\frac{x^2-(25)^2}{x-(25)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpre... | [
{
"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: $\\boxed{50}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=50$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 50$. |
math-002409 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Find the exact value: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 318}\frac{x^2-(318)^2}{x-(318)}.$$
(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: $x^2-(318)^2=(x-(318))(x+(318))$.",
"Step 2: For $x\\neq 318$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{636}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=636$. 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 $2a= 636$. |
math-002410 | Calculus: Limits — Algebraic Simplification | 2 | 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:
$$\lim_{x\to 394}\frac{x^2-(394)^2}{x-(394)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{788}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=788$. 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 $2a= 788$. (Here the result is $\boxed{788}$.) |
math-002411 | Calculus: Limits — Removable Discontinuities | 2 | Solve (and briefly cross-validate): 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:
$$\lim_{x\to -109}\frac{x^2-(-109)^2}{x-(-109)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-218}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-218$. 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 $2a= -218$. |
math-002412 | Calculus: Limits — Algebraic Simplification | 2 | 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:
$$\lim_{x\to -145}\frac{x^2-(-145)^2}{x-(-145)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinter... | [
{
"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: $\\boxed{-290}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-290$. 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 $2a= -290$. (Here the result is $\boxed{-290}$.) |
math-002413 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to -128}\frac{x^2-(-128)^2}{x-(-128)}.$$
(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: $x^2-(-128)^2=(x-(-128))(x+(-128))$.",
"Step 2: For $x\\neq -128$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. 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 $a$.",
"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 $2a= -256$. |
math-002414 | Calculus: Limits — Removable Discontinuities | 2 | Carefully track domains: 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:
$$\lim_{x\to 395}\frac{x^2-(395)^2}{x-(395)}.$$
(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: $x^2-(395)^2=(x-(395))(x+(395))$.",
"Step 2: For $x\\neq 395$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. 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 $a$.",
"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 $2a= 790$. |
math-002415 | Calculus: Limits — Difference Quotients | 2 | Indicate where a theorem is used: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 187}\frac{x^2-(187)^2}{x-(187)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivativ... | [
{
"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: $\\boxed{374}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=374$. 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 $2a= 374$. (Here the result is $\boxed{374}$.) |
math-002416 | Calculus: Limits — Removable Discontinuities | 2 | Provide a rigorous 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:
$$\lim_{x\to -243}\frac{x^2-(-243)^2}{x-(-243)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Re... | [
{
"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: $\\boxed{-486}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-486$. 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 $2a= -486$. (Here the result is $\boxed{-486}$.) |
math-002417 | Calculus: Limits — Removable Discontinuities | 2 | 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:
$$\lim_{x\to -302}\frac{x^2-(-302)^2}{x-(-302)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and com... | [
{
"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: $x^2-(-302)^2=(x-(-302))(x+(-302))$.",
"Step 2: For $x\\neq -302$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. 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 $a$.",
"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 $2a= -604$. (Here the result is $\boxed{-604}$.) |
math-002418 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to 299}\frac{x^2-(299)^2}{x-(299)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{598}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=598$. 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 $2a= 598$. (Here the result is $\boxed{598}$.) |
math-002419 | Calculus: Limits — Removable Discontinuities | 2 | Give a theorem-based solution: 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:
$$\lim_{x\to -271}\frac{x^2-(-271)^2}{x-(-271)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret t... | [
{
"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: $\\boxed{-542}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-542$. 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... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -542$. |
math-002420 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Prompt: 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:
$$\lim_{x\to 264}\frac{x^2-(264)^2}{x-(264)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit... | [
{
"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: $x^2-(264)^2=(x-(264))(x+(264))$.",
"Step 2: For $x\\neq 264$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{528}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=528$. 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 $2a= 528$. |
math-002421 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Explain why your operations are valid: 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:
$$\lim_{x\to -94}\frac{x^2-(-94)^2}{x-(-94)}.$$
(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: $x^2-(-94)^2=(x-(-94))(x+(-94))$.",
"Step 2: For $x\\neq -94$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-188}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-188$. 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 $2a= -188$. (Here the result is $\boxed{-188}$.) |
math-002422 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Carefully track domains: 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:
$$\lim_{x\to 42}\frac{x^2-(42)^2}{x-(42)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"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: $x^2-(42)^2=(x-(42))(x+(42))$.",
"Step 2: For $x\\neq 42$, cancel to get $\\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{84}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=84$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analy... | [
{
"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 $2a= 84$. (Here the result is $\boxed{84}$.) |
math-002423 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to -91}\frac{x^2-(-91)^2}{x-(-91)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative a... | [
{
"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: $\\boxed{-182}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-182$. 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 $2a= -182$. (Here the result is $\boxed{-182}$.) |
math-002424 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Work this out 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:
$$\lim_{x\to -138}\frac{x^2-(-138)^2}{x-(-138)}.$$
(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: $x^2-(-138)^2=(x-(-138))(x+(-138))$.",
"Step 2: For $x\\neq -138$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-276}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-276$. 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 $2a= -276$. |
math-002425 | Calculus: Limits — Removable Discontinuities | 2 | Give an answer and a quick verification: 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:
$$\lim_{x\to 368}\frac{x^2-(368)^2}{x-(368)}.$$
(a) Evaluate the limit by algebraic simplifi... | [
{
"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: $\\boxed{736}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=736$. 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... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 736$. (Here the result is $\boxed{736}$.) |
math-002426 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Explain why your operations are valid: 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:
$$\lim_{x\to -210}\frac{x^2-(-210)^2}{x-(-210)}.$$
(a) Evaluate the limit by algebraic simplif... | [
{
"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: $x^2-(-210)^2=(x-(-210))(x+(-210))$.",
"Step 2: For $x\\neq -210$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-420}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-420$. 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... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -420$. (Here the result is $\boxed{-420}$.) |
math-002427 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Solve and include a self-check: 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:
$$\lim_{x\to -328}\frac{x^2-(-328)^2}{x-(-328)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"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: $x^2-(-328)^2=(x-(-328))(x+(-328))$.",
"Step 2: For $x\\neq -328$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. 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": "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 $2a= -656$. |
math-002428 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to -223}\frac{x^2-(-223)^2}{x-(-223)}.$$
(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: $\\boxed{-446}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-446$. 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... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -446$. |
math-002429 | Calculus: Limits — Removable Discontinuities | 2 | 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:
$$\lim_{x\to 225}\frac{x^2-(225)^2}{x-(225)}.$$
(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: $\\boxed{450}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=450$. 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 $2a= 450$. (Here the result is $\boxed{450}$.) |
math-002430 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to -113}\frac{x^2-(-113)^2}{x-(-113)}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-226}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-226$. 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 $2a= -226$. (Here the result is $\boxed{-226}$.) |
math-002431 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to -57}\frac{x^2-(-57)^2}{x-(-57)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the lim... | [
{
"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: $\\boxed{-114}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-114$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -114$. (Here the result is $\boxed{-114}$.) |
math-002432 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Do not skip justification steps: 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:
$$\lim_{x\to -339}\frac{x^2-(-339)^2}{x-(-339)}.$$
(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: $x^2-(-339)^2=(x-(-339))(x+(-339))$.",
"Step 2: For $x\\neq -339$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-678}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-678$. 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 $2a= -678$. |
math-002433 | Calculus: Limits — Algebraic Simplification | 2 | Checkpoint: 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:
$$\lim_{x\to 203}\frac{x^2-(203)^2}{x-(203)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the l... | [
{
"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: $\\boxed{406}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=406$. 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 $2a= 406$. |
math-002434 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to 110}\frac{x^2-(110)^2}{x-(110)}.$$
(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: $\\boxed{220}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=220$. 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 $2a= 220$. (Here the result is $\boxed{220}$.) |
math-002435 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to -221}\frac{x^2-(-221)^2}{x-(-221)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and co... | [
{
"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: $\\boxed{-442}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-442$. 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 $2a= -442$. |
math-002436 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to 117}\frac{x^2-(117)^2}{x-(117)}.$$
(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: $\\boxed{234}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=234$. 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 $2a= 234$. (Here the result is $\boxed{234}$.) |
math-002437 | Calculus: Limits — Removable Discontinuities | 2 | 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:
$$\lim_{x\to -241}\frac{x^2-(-241)^2}{x-(-241)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-482}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-482$. 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 $2a= -482$. |
math-002438 | Calculus: Limits — Removable Discontinuities | 2 | Start by stating any domain restrictions: 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:
$$\lim_{x\to -163}\frac{x^2-(-163)^2}{x-(-163)}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-326}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-326$. 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 $2a= -326$. |
math-002439 | Calculus: Limits — Difference Quotients | 2 | Give reasoning, not just computation: 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:
$$\lim_{x\to -83}\frac{x^2-(-83)^2}{x-(-83)}.$$
(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: $x^2-(-83)^2=(x-(-83))(x+(-83))$.",
"Step 2: For $x\\neq -83$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-166}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-166$. 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 $2a= -166$. |
math-002440 | Calculus: Limits — Difference Quotients | 2 | Determine the requested value: 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:
$$\lim_{x\to -161}\frac{x^2-(-161)^2}{x-(-161)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret t... | [
{
"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: $\\boxed{-322}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-322$. 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... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -322$. (Here the result is $\boxed{-322}$.) |
math-002441 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Write the solution set clearly: 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:
$$\lim_{x\to 107}\frac{x^2-(107)^2}{x-(107)}.$$
(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: $x^2-(107)^2=(x-(107))(x+(107))$.",
"Step 2: For $x\\neq 107$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{214}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=214$. 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 $2a= 214$. (Here the result is $\boxed{214}$.) |
math-002442 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Solve with verification: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 112}\frac{x^2-(112)^2}{x-(112)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and com... | [
{
"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: $\\boxed{224}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=224$. 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 $2a= 224$. |
math-002443 | Calculus: Limits — Difference Quotients | 2 | Write the solution set clearly: 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:
$$\lim_{x\to 171}\frac{x^2-(171)^2}{x-(171)}.$$
(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: $\\boxed{342}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=342$. 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 $2a= 342$. (Here the result is $\boxed{342}$.) |
math-002444 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to 30}\frac{x^2-(30)^2}{x-(30)}.$$
(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: $\\boxed{60}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=60$. 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... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 60$. |
math-002445 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to -279}\frac{x^2-(-279)^2}{x-(-279)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Re... | [
{
"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: $\\boxed{-558}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-558$. 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 $2a= -558$. (Here the result is $\boxed{-558}$.) |
math-002446 | Calculus: Limits — Algebraic Simplification | 2 | Task: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -226}\frac{x^2-(-226)^2}{x-(-226)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that way... | [
{
"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: $\\boxed{-452}$.\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_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 $2a= -452$. (Here the result is $\boxed{-452}$.) |
math-002447 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Carefully track domains: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 118}\frac{x^2-(118)^2}{x-(118)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and com... | [
{
"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: $x^2-(118)^2=(x-(118))(x+(118))$.",
"Step 2: For $x\\neq 118$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{236}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=236$. 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 $2a= 236$. (Here the result is $\boxed{236}$.) |
math-002448 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Answer using clear logical steps: 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:
$$\lim_{x\to 205}\frac{x^2-(205)^2}{x-(205)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret t... | [
{
"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: $x^2-(205)^2=(x-(205))(x+(205))$.",
"Step 2: For $x\\neq 205$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{410}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=410$. 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 $2a= 410$. (Here the result is $\boxed{410}$.) |
math-002449 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Where appropriate, name the theorem you use: 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:
$$\lim_{x\to -187}\frac{x^2-(-187)^2}{x-(-187)}.$$
(a) Evaluate the limit by algebraic simpli... | [
{
"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: $x^2-(-187)^2=(x-(-187))(x+(-187))$.",
"Step 2: For $x\\neq -187$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-374}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-374$. 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 $2a= -374$. |
math-002450 | Calculus: Limits — Algebraic Simplification | 2 | Make each step logically reversible (or explain if not): 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:
$$\lim_{x\to 272}\frac{x^2-(272)^2}{x-(272)}.$$
(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: $\\boxed{544}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=544$. 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 $2a= 544$. (Here the result is $\boxed{544}$.) |
math-002451 | Calculus: Limits — Removable Discontinuities | 2 | Work carefully and justify each inference: 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:
$$\lim_{x\to 240}\frac{x^2-(240)^2}{x-(240)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{480}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=480$. 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 $2a= 480$. |
math-002452 | Calculus: Limits — Removable Discontinuities | 2 | Work carefully and justify each inference: 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:
$$\lim_{x\to 213}\frac{x^2-(213)^2}{x-(213)}.$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{426}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=426$. 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 $2a= 426$. (Here the result is $\boxed{426}$.) |
math-002453 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Provide both a computational and a conceptual explanation: 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:
$$\lim_{x\to 204}\frac{x^2-(204)^2}{x-(204)}.$$
(a) Evaluate the limit by ... | [
{
"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: $\\boxed{408}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=408$. 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 $2a= 408$. |
math-002454 | Calculus: Limits — Algebraic Simplification | 2 | Work this out 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:
$$\lim_{x\to -134}\frac{x^2-(-134)^2}{x-(-134)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"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: $\\boxed{-268}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-268$. 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 $2a= -268$. |
math-002455 | Calculus: Limits — Difference Quotients | 2 | Solve and include a self-check: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -245}\frac{x^2-(-245)^2}{x-(-245)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivati... | [
{
"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: $\\boxed{-490}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-490$. 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 $2a= -490$. (Here the result is $\boxed{-490}$.) |
math-002456 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to -366}\frac{x^2-(-366)^2}{x-(-366)}.$$
(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: $\\boxed{-732}$.\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 $a$.",
"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 $2a= -732$. (Here the result is $\boxed{-732}$.) |
math-002457 | Calculus: Limits — Difference Quotients | 2 | Explain what is being counted/optimized: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -269}\frac{x^2-(-269)^2}{x-(-269)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a... | [
{
"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: $\\boxed{-538}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-538$. 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 $2a= -538$. (Here the result is $\boxed{-538}$.) |
math-002458 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to 378}\frac{x^2-(378)^2}{x-(378)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{756}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=756$. 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 $2a= 756$. (Here the result is $\boxed{756}$.) |
math-002459 | Calculus: Limits — Algebraic Simplification | 2 | Explain why your operations are valid: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 391}\frac{x^2-(391)^2}{x-(391)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a deri... | [
{
"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: $x^2-(391)^2=(x-(391))(x+(391))$.",
"Step 2: For $x\\neq 391$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{782}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=782$. 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 $2a= 782$. |
math-002460 | Calculus: Limits — Algebraic Simplification | 2 | Find the exact value: 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:
$$\lim_{x\to -203}\frac{x^2-(-203)^2}{x-(-203)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpr... | [
{
"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: $\\boxed{-406}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-406$. 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 $2a= -406$. |
math-002461 | Calculus: Limits — Removable Discontinuities | 2 | Work carefully and justify each inference: 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:
$$\lim_{x\to 383}\frac{x^2-(383)^2}{x-(383)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rein... | [
{
"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: $\\boxed{766}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=766$. 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 $2a= 766$. (Here the result is $\boxed{766}$.) |
math-002462 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Prompt: 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:
$$\lim_{x\to -387}\frac{x^2-(-387)^2}{x-(-387)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the li... | [
{
"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: $x^2-(-387)^2=(x-(-387))(x+(-387))$.",
"Step 2: For $x\\neq -387$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-774}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-774$. 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 $2a= -774$. |
math-002463 | Calculus: Limits — Difference Quotients | 2 | Indicate where a theorem is used: 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:
$$\lim_{x\to 336}\frac{x^2-(336)^2}{x-(336)}.$$
(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: $\\boxed{672}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=672$. 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 $2a= 672$. (Here the result is $\boxed{672}$.) |
math-002464 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to 197}\frac{x^2-(197)^2}{x-(197)}.$$
(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: $x^2-(197)^2=(x-(197))(x+(197))$.",
"Step 2: For $x\\neq 197$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{394}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=394$. 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 $2a= 394$. (Here the result is $\boxed{394}$.) |
math-002465 | Calculus: Limits — Difference Quotients | 2 | Checkpoint: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -99}\frac{x^2-(-99)^2}{x-(-99)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that ... | [
{
"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: $\\boxed{-198}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-198$. 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... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -198$. (Here the result is $\boxed{-198}$.) |
math-002466 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to 72}\frac{x^2-(72)^2}{x-(72)}.$$
(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: $\\boxed{144}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=144$. 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 $2a= 144$. (Here the result is $\boxed{144}$.) |
math-002467 | Calculus: Limits — Difference Quotients | 2 | Answer with a short justification: 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:
$$\lim_{x\to 142}\frac{x^2-(142)^2}{x-(142)}.$$
(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: $x^2-(142)^2=(x-(142))(x+(142))$.",
"Step 2: For $x\\neq 142$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{284}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=284$. 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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 284$. (Here the result is $\boxed{284}$.) |
math-002468 | Calculus: Limits — Removable Discontinuities | 2 | 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:
$$\lim_{x\to 371}\frac{x^2-(371)^2}{x-(371)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{742}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=742$. 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 $2a= 742$. |
math-002469 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Work carefully and justify each inference: 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:
$$\lim_{x\to 152}\frac{x^2-(152)^2}{x-(152)}.$$
(a) Evaluate the limit by algebraic simplificat... | [
{
"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: $\\boxed{304}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=304$. 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 $2a= 304$. (Here the result is $\boxed{304}$.) |
math-002470 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Checkpoint: 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:
$$\lim_{x\to -66}\frac{x^2-(-66)^2}{x-(-66)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivati... | [
{
"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: $\\boxed{-132}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-132$. 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... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -132$. (Here the result is $\boxed{-132}$.) |
math-002471 | Calculus: Limits — Removable Discontinuities | 2 | Solve and sanity-check: 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:
$$\lim_{x\to -190}\frac{x^2-(-190)^2}{x-(-190)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limi... | [
{
"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: $\\boxed{-380}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-380$. 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 $2a= -380$. (Here the result is $\boxed{-380}$.) |
math-002472 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Indicate where a theorem is used: 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:
$$\lim_{x\to 154}\frac{x^2-(154)^2}{x-(154)}.$$
(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: $x^2-(154)^2=(x-(154))(x+(154))$.",
"Step 2: For $x\\neq 154$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{308}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=308$. 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 $2a= 308$. (Here the result is $\boxed{308}$.) |
math-002473 | Calculus: Limits — Difference Quotients | 2 | Derive the result step-by-step: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 312}\frac{x^2-(312)^2}{x-(312)}.$$
(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": "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": "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 $2a= 624$. (Here the result is $\boxed{624}$.) |
math-002474 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to 68}\frac{x^2-(68)^2}{x-(68)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and... | [
{
"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: $\\boxed{136}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=136$. 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 $2a= 136$. |
math-002475 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to -56}\frac{x^2-(-56)^2}{x-(-56)}.$$
(a) Evaluate the limit by algebraic simplifi... | [
{
"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: $\\boxed{-112}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-112$. 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 $2a= -112$. (Here the result is $\boxed{-112}$.) |
math-002476 | Calculus: Limits — Removable Discontinuities | 2 | Checkpoint: 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:
$$\lim_{x\to 82}\frac{x^2-(82)^2}{x-(82)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{164}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=164$. 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 $2a= 164$. (Here the result is $\boxed{164}$.) |
math-002477 | Calculus: Limits — Removable Discontinuities | 2 | 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:
$$\lim_{x\to -399}\frac{x^2-(-399)^2}{x-(-399)}.$$
(a) Evaluate the limit by algebraic simp... | [
{
"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: $\\boxed{-798}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-798$. 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 $2a= -798$. |
math-002478 | Calculus: Limits — Algebraic Simplification | 2 | Compute the requested quantity: 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:
$$\lim_{x\to -233}\frac{x^2-(-233)^2}{x-(-233)}.$$
(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: $\\boxed{-466}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-466$. 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 $2a= -466$. (Here the result is $\boxed{-466}$.) |
math-002479 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to 15}\frac{x^2-(15)^2}{x-(15)}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{30}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=30$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitivit... | [
{
"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 $2a= 30$. (Here the result is $\boxed{30}$.) |
math-002480 | Calculus: Limits — Removable Discontinuities | 2 | 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:
$$\lim_{x\to -62}\frac{x^2-(-62)^2}{x-(-62)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-124}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-124$. 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 $2a= -124$. |
math-002481 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Find the exact value: 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:
$$\lim_{x\to -69}\frac{x^2-(-69)^2}{x-(-69)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as ... | [
{
"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: $\\boxed{-138}$.\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_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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -138$. (Here the result is $\boxed{-138}$.) |
math-002482 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Challenge: 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:
$$\lim_{x\to 135}\frac{x^2-(135)^2}{x-(135)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit a... | [
{
"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: $\\boxed{270}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=270$. 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 $2a= 270$. (Here the result is $\boxed{270}$.) |
math-002483 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Task: 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:
$$\lim_{x\to -268}\frac{x^2-(-268)^2}{x-(-268)}.$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{-536}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-536$. 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 $2a= -536$. |
math-002484 | Calculus: Limits — Removable Discontinuities | 2 | 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:
$$\lim_{x\to -289}\frac{x^2-(-289)^2}{x-(-289)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-578}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-578$. 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 $2a= -578$. (Here the result is $\boxed{-578}$.) |
math-002485 | Calculus: Limits — Algebraic Simplification | 2 | Question: 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:
$$\lim_{x\to -355}\frac{x^2-(-355)^2}{x-(-355)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-710}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-710$. 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 $2a= -710$. |
math-002486 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Give a theorem-based solution: 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:
$$\lim_{x\to 388}\frac{x^2-(388)^2}{x-(388)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{776}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=776$. 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 $2a= 776$. (Here the result is $\boxed{776}$.) |
math-002487 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Solve (and briefly cross-validate): Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 348}\frac{x^2-(348)^2}{x-(348)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivat... | [
{
"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: $\\boxed{696}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=696$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 696$. (Here the result is $\boxed{696}$.) |
math-002488 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Answer with a short justification: 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:
$$\lim_{x\to -337}\frac{x^2-(-337)^2}{x-(-337)}.$$
(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: $\\boxed{-674}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-674$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -674$. |
math-002489 | Calculus: Limits — Removable Discontinuities | 2 | Write the solution set clearly: 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:
$$\lim_{x\to -287}\frac{x^2-(-287)^2}{x-(-287)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"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: $\\boxed{-574}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-574$. 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 $2a= -574$. (Here the result is $\boxed{-574}$.) |
math-002490 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to -23}\frac{x^2-(-23)^2}{x-(-23)}.$$
(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: $x^2-(-23)^2=(x-(-23))(x+(-23))$.",
"Step 2: For $x\\neq -23$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-46}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-46$. 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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -46$. |
math-002491 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Try to avoid pattern-matching; explain why: 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:
$$\lim_{x\to -272}\frac{x^2-(-272)^2}{x-(-272)}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-544}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-544$. 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... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -544$. (Here the result is $\boxed{-544}$.) |
math-002492 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Determine the requested value: 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:
$$\lim_{x\to 113}\frac{x^2-(113)^2}{x-(113)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{226}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=226$. 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 $2a= 226$. |
math-002493 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Write the solution set clearly: 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:
$$\lim_{x\to 289}\frac{x^2-(289)^2}{x-(289)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Re... | [
{
"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: $\\boxed{578}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=578$. 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 $2a= 578$. (Here the result is $\boxed{578}$.) |
math-002494 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to 322}\frac{x^2-(322)^2}{x-(322)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{644}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=644$. 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 $2a= 644$. (Here the result is $\boxed{644}$.) |
math-002495 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Make each step logically reversible (or explain if not): Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -222}\frac{x^2-(-222)^2}{x-(-222)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpre... | [
{
"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: $\\boxed{-444}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-444$. 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 $2a= -444$. (Here the result is $\boxed{-444}$.) |
math-002496 | Calculus: Limits — Difference Quotients | 2 | Warm-up: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -105}\frac{x^2-(-105)^2}{x-(-105)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that ... | [
{
"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: $\\boxed{-210}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-210$. 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 $2a= -210$. (Here the result is $\boxed{-210}$.) |
math-002497 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Solve and justify each step: 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:
$$\lim_{x\to -82}\frac{x^2-(-82)^2}{x-(-82)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-164}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-164$. 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 $2a= -164$. (Here the result is $\boxed{-164}$.) |
math-002498 | Calculus: Limits — Removable Discontinuities | 2 | Question: 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:
$$\lim_{x\to 22}\frac{x^2-(22)^2}{x-(22)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a ... | [
{
"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: $\\boxed{44}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=44$. 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 $2a= 44$. (Here the result is $\boxed{44}$.) |
math-002499 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to 21}\frac{x^2-(21)^2}{x-(21)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{42}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=42$. 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 $2a= 42$. |
math-002500 | Calculus: Limits — Difference Quotients | 2 | Proceed methodically: 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:
$$\lim_{x\to 370}\frac{x^2-(370)^2}{x-(370)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"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: $\\boxed{740}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=740$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 740$. |
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