id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-000601 | Calculus: Limits — Algebraic Simplification | 1 | Provide both a computational and a conceptual explanation: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 45}\frac{x^2-(45)^2}{x-(45)}.$$
(a) Evaluate the limit by algebraic simplificat... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{90}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=90$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 90$. (Here the result is $\boxed{90}$.) |
math-000602 | Calculus: Limits — Difference Quotients | 1 | Explain what is being counted/optimized: 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:
$$\lim_{x\to -81}\frac{x^2-(-81)^2}{x-(-81)}.$$
(a) Evaluate the limit by algebraic simplificatio... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-162}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-162$. 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 $2a= -162$. (Here the result is $\boxed{-162}$.) |
math-000603 | Calculus: Limits — Difference Quotients | 1 | Problem: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -55}\frac{x^2-(-55)^2}{x-(-55)}.$$
(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: $\\boxed{-110}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-110$. 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 $2a= -110$. |
math-000604 | Calculus: Limits — Algebraic Simplification | 1 | 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:
$$\lim_{x\to -80}\frac{x^2-(-80)^2}{x-(-80)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a d... | [
{
"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: $\\boxed{-160}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-160$. 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 $2a= -160$. (Here the result is $\boxed{-160}$.) |
math-000605 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to 214}\frac{x^2-(214)^2}{x-(214)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinter... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(214)^2=(x-(214))(x+(214))$.",
"Step 2: For $x\\neq 214$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{428}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=428$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the p... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 428$. (Here the result is $\boxed{428}$.) |
math-000606 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | 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:
$$\lim_{x\to -326}\frac{x^2-(-326)^2}{x-(-326)}.$$
(a) Evaluate the limit by algebraic simplificati... | [
{
"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: $\\boxed{-652}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-652$. 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 $2a= -652$. (Here the result is $\boxed{-652}$.) |
math-000607 | Calculus: Limits — Algebraic Simplification | 1 | Solve and include a self-check: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 47}\frac{x^2-(47)^2}{x-(47)}.$$
(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: $x^2-(47)^2=(x-(47))(x+(47))$.",
"Step 2: For $x\\neq 47$, cancel to get $\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{94}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=94$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustn... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 94$. (Here the result is $\boxed{94}$.) |
math-000608 | Calculus: Limits — Algebraic Simplification | 1 | State any required conditions first: 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:
$$\lim_{x\to 33}\frac{x^2-(33)^2}{x-(33)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret t... | [
{
"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: $\\boxed{66}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=66$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality... | [
{
"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 $2a= 66$. |
math-000609 | Calculus: Limits — Difference Quotients | 1 | Derive the result step-by-step: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 219}\frac{x^2-(219)^2}{x-(219)}.$$
(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: $x^2-(219)^2=(x-(219))(x+(219))$.",
"Step 2: For $x\\neq 219$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{438}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=438$. 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 $2a= 438$. (Here the result is $\boxed{438}$.) |
math-000610 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Prompt: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 297}\frac{x^2-(297)^2}{x-(297)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{594}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=594$. 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 $2a= 594$. |
math-000611 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Track units/moduli carefully: 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:
$$\lim_{x\to 259}\frac{x^2-(259)^2}{x-(259)}.$$
(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: $\\boxed{518}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=518$. 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 $2a= 518$. |
math-000612 | Calculus: Limits — Removable Discontinuities | 1 | Proceed methodically: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -117}\frac{x^2-(-117)^2}{x-(-117)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-234}$.\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": "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 $2a= -234$. (Here the result is $\boxed{-234}$.) |
math-000613 | Calculus: Limits — Difference Quotients | 1 | Solve and then verify: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 355}\frac{x^2-(355)^2}{x-(355)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compu... | [
{
"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: $x^2-(355)^2=(x-(355))(x+(355))$.",
"Step 2: For $x\\neq 355$, cancel to get... | {
"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_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 $2a= 710$. |
math-000614 | Calculus: Limits — Removable Discontinuities | 1 | Problem: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -172}\frac{x^2-(-172)^2}{x-(-172)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) 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: $x^2-(-172)^2=(x-(-172))(x+(-172))$.",
"Step 2: For $x\\neq -172$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-344}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-344$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -344$. (Here the result is $\boxed{-344}$.) |
math-000615 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Provide a rigorous solution: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 232}\frac{x^2-(232)^2}{x-(232)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reint... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{464}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=464$. 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 $2a= 464$. |
math-000616 | Calculus: Limits — Algebraic Simplification | 1 | Compute the requested quantity: 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:
$$\lim_{x\to -150}\frac{x^2-(-150)^2}{x-(-150)}.$$
(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: $x^2-(-150)^2=(x-(-150))(x+(-150))$.",
"Step 2: For $x\\neq -150$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-300}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-300$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -300$. (Here the result is $\boxed{-300}$.) |
math-000617 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Give a theorem-based solution: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 338}\frac{x^2-(338)^2}{x-(338)}.$$
(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: $\\boxed{676}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=676$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= 676$. |
math-000618 | Calculus: Limits — Removable Discontinuities | 1 | Solve (and briefly cross-validate): Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 70}\frac{x^2-(70)^2}{x-(70)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(70)^2=(x-(70))(x+(70))$.",
"Step 2: For $x\\neq 70$, cancel to get $\\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{140}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=140$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitiv... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 140$. |
math-000619 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Use two approaches if possible: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 31}\frac{x^2-(31)^2}{x-(31)}.$$
(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: $x^2-(31)^2=(x-(31))(x+(31))$.",
"Step 2: For $x\\neq 31$, cancel to get $\\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{62}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=62$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality note:... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 62$. (Here the result is $\boxed{62}$.) |
math-000620 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Explain what is being counted/optimized: 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:
$$\lim_{x\to 95}\frac{x^2-(95)^2}{x-(95)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpr... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{190}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=190$. 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 $2a= 190$. |
math-000621 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | 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:
$$\lim_{x\to 138}\frac{x^2-(138)^2}{x-(138)}.$$
(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": "Both approaches agree after simplification. 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 $a$.",
"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 $2a= 276$. (Here the result is $\boxed{276}$.) |
math-000622 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Solve and justify each step: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -141}\frac{x^2-(-141)^2}{x-(-141)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the... | [
{
"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: $x^2-(-141)^2=(x-(-141))(x+(-141))$.",
"Step 2: For $x\\neq -141$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-282}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-282$. 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 $2a= -282$. |
math-000623 | Calculus: Limits — Removable Discontinuities | 1 | Answer with a short justification: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -247}\frac{x^2-(-247)^2}{x-(-247)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a deriv... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-247)^2=(x-(-247))(x+(-247))$.",
"Step 2: For $x\\neq -247$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-494}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-494$. 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 $2a= -494$. |
math-000624 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Make each step logically reversible (or explain if not): 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:
$$\lim_{x\to 1}\frac{x^2-(1)^2}{x-(1)}.$$
(a) Evaluate the limit by algebraic sim... | [
{
"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: $x^2-(1)^2=(x-(1))(x+(1))$.",
"Step 2: For $x\\neq 1$, cancel to get $\\frac... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{2}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=2$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysi... | [
{
"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 $2a= 2$. (Here the result is $\boxed{2}$.) |
math-000625 | Calculus: Limits — Removable Discontinuities | 1 | Solve with verification: 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:
$$\lim_{x\to -84}\frac{x^2-(-84)^2}{x-(-84)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpr... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-168}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-168$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -168$. (Here the result is $\boxed{-168}$.) |
math-000626 | Calculus: Limits — Removable Discontinuities | 1 | Solve and justify each step: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -159}\frac{x^2-(-159)^2}{x-(-159)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Re... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-159)^2=(x-(-159))(x+(-159))$.",
"Step 2: For $x\\neq -159$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-318}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-318$. 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 $2a= -318$. (Here the result is $\boxed{-318}$.) |
math-000627 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Solve and include a self-check: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 6}\frac{x^2-(6)^2}{x-(6)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{12}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=12$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem w... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 12$. |
math-000628 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Prompt: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 97}\frac{x^2-(97)^2}{x-(97)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a de... | [
{
"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: $\\boxed{194}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=194$. 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 $2a= 194$. |
math-000629 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Problem: 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:
$$\lim_{x\to -333}\frac{x^2-(-333)^2}{x-(-333)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-666}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-666$. 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 $2a= -666$. (Here the result is $\boxed{-666}$.) |
math-000630 | Calculus: Limits — Difference Quotients | 1 | Write the solution set clearly: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 293}\frac{x^2-(293)^2}{x-(293)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative ... | [
{
"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: $x^2-(293)^2=(x-(293))(x+(293))$.",
"Step 2: For $x\\neq 293$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{586}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=586$. 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 $2a= 586$. |
math-000631 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Try to avoid pattern-matching; explain why: 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:
$$\lim_{x\to -371}\frac{x^2-(-371)^2}{x-(-371)}.$$
(a) Evaluate the limit by algebraic simplif... | [
{
"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: $\\boxed{-742}$.\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 $a$.",
"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 $2a= -742$. |
math-000632 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Explain what is being counted/optimized: 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:
$$\lim_{x\to 246}\frac{x^2-(246)^2}{x-(246)}.$$
(a) Evaluate the limit by algebraic simplificatio... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(246)^2=(x-(246))(x+(246))$.",
"Step 2: For $x\\neq 246$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{492}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=492$. 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 $2a= 492$. |
math-000633 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Track units/moduli carefully: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -71}\frac{x^2-(-71)^2}{x-(-71)}.$$
(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: $\\boxed{-142}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-142$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -142$. (Here the result is $\boxed{-142}$.) |
math-000634 | Calculus: Limits — Difference Quotients | 1 | Track quantifiers carefully: 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:
$$\lim_{x\to -373}\frac{x^2-(-373)^2}{x-(-373)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Re... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-373)^2=(x-(-373))(x+(-373))$.",
"Step 2: For $x\\neq -373$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-746}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-746$. 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 $2a= -746$. |
math-000635 | Calculus: Limits — Removable Discontinuities | 1 | Start by stating any domain restrictions: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 273}\frac{x^2-(273)^2}{x-(273)}.$$
(a) Evaluate the limit by algebraic simplificati... | [
{
"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: $x^2-(273)^2=(x-(273))(x+(273))$.",
"Step 2: For $x\\neq 273$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{546}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=546$. 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 $2a= 546$. (Here the result is $\boxed{546}$.) |
math-000636 | Calculus: Limits — Removable Discontinuities | 1 | Keep the final answer in boxed form: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 282}\frac{x^2-(282)^2}{x-(282)}.$$
(a) Evaluate the limit by algebraic simplificati... | [
{
"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: $\\boxed{564}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=564$. 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 $2a= 564$. (Here the result is $\boxed{564}$.) |
math-000637 | Calculus: Limits — Removable Discontinuities | 1 | Start by stating any domain restrictions: 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 d... | [
{
"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: $\\boxed{-70}$.\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_analysis": "Robustne... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -70$. |
math-000638 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Explain what is being counted/optimized: 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:
$$\lim_{x\to -394}\frac{x^2-(-394)^2}{x-(-394)}.$$
(a) Evaluate the limit by algebraic simpl... | [
{
"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: $x^2-(-394)^2=(x-(-394))(x+(-394))$.",
"Step 2: For $x\\neq -394$, cancel to... | {
"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_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 $2a= -788$. (Here the result is $\boxed{-788}$.) |
math-000639 | Calculus: Limits — Algebraic Simplification | 1 | 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:
$$\lim_{x\to 49}\frac{x^2-(49)^2}{x-(49)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{98}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=98$. 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 $2a= 98$. |
math-000640 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Where appropriate, name the theorem you use: 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:
$$\lim_{x\to -357}\frac{x^2-(-357)^2}{x-(-357)}.$$
(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: $x^2-(-357)^2=(x-(-357))(x+(-357))$.",
"Step 2: For $x\\neq -357$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-714}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-714$. 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... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -714$. |
math-000641 | Calculus: Limits — Difference Quotients | 1 | Solve and justify each step: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 56}\frac{x^2-(56)^2}{x-(56)}.$$
(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: $x^2-(56)^2=(x-(56))(x+(56))$.",
"Step 2: For $x\\neq 56$, cancel to get $\\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. 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 $a$.",
"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 $2a= 112$. (Here the result is $\boxed{112}$.) |
math-000642 | Calculus: Limits — Difference Quotients | 1 | Keep the final answer in boxed form: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -249}\frac{x^2-(-249)^2}{x-(-249)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-498}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-498$. 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 $2a= -498$. |
math-000643 | Calculus: Limits — Removable Discontinuities | 1 | Determine the requested 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:
$$\lim_{x\to -317}\frac{x^2-(-317)^2}{x-(-317)}.$$
(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: $x^2-(-317)^2=(x-(-317))(x+(-317))$.",
"Step 2: For $x\\neq -317$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-634}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-634$. 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 $2a= -634$. (Here the result is $\boxed{-634}$.) |
math-000644 | Calculus: Limits — Algebraic Simplification | 1 | Solve and sanity-check: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -238}\frac{x^2-(-238)^2}{x-(-238)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and c... | [
{
"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: $x^2-(-238)^2=(x-(-238))(x+(-238))$.",
"Step 2: For $x\\neq -238$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-476}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-476$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -476$. (Here the result is $\boxed{-476}$.) |
math-000645 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Give a theorem-based solution: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -392}\frac{x^2-(-392)^2}{x-(-392)}.$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{-784}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-784$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -784$. |
math-000646 | Calculus: Limits — Algebraic Simplification | 1 | Problem: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 278}\frac{x^2-(278)^2}{x-(278)}.$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{556}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=556$. 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 $2a= 556$. (Here the result is $\boxed{556}$.) |
math-000647 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Indicate where a theorem is used: 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:
$$\lim_{x\to -318}\frac{x^2-(-318)^2}{x-(-318)}.$$
(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: $\\boxed{-636}$.\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_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 $2a= -636$. |
math-000648 | Calculus: Limits — Removable Discontinuities | 1 | Solve (and briefly cross-validate): Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -9}\frac{x^2-(-9)^2}{x-(-9)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-18}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-18$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -18$. (Here the result is $\boxed{-18}$.) |
math-000649 | Calculus: Limits — Algebraic Simplification | 1 | Give a fully justified solution: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -160}\frac{x^2-(-160)^2}{x-(-160)}.$$
(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: $x^2-(-160)^2=(x-(-160))(x+(-160))$.",
"Step 2: For $x\\neq -160$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-320}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-320$. 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 $2a= -320$. |
math-000650 | Calculus: Limits — Removable Discontinuities | 1 | 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:
$$\lim_{x\to 367}\frac{x^2-(367)^2}{x-(367)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{734}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=734$. 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 $2a= 734$. (Here the result is $\boxed{734}$.) |
math-000651 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to -320}\frac{x^2-(-320)^2}{x-(-320)}.$$
(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: $\\boxed{-640}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-640$. 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 $2a= -640$. (Here the result is $\boxed{-640}$.) |
math-000652 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Do not skip justification steps: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -393}\frac{x^2-(-393)^2}{x-(-393)}.$$
(a) Evaluate the limit by algebraic simplification.
(b... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-786}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-786$. 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 $2a= -786$. (Here the result is $\boxed{-786}$.) |
math-000653 | Calculus: Limits — Algebraic Simplification | 1 | 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:
$$\lim_{x\to 3}\frac{x^2-(3)^2}{x-(3)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{6}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=6$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustnes... | [
{
"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 $2a= 6$. (Here the result is $\boxed{6}$.) |
math-000654 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Prompt: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -231}\frac{x^2-(-231)^2}{x-(-231)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-462}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-462$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -462$. |
math-000655 | Calculus: Limits — Algebraic Simplification | 1 | Prompt: 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:
$$\lim_{x\to 153}\frac{x^2-(153)^2}{x-(153)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative 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: $\\boxed{306}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=306$. 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 $2a= 306$. |
math-000656 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | 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:
$$\lim_{x\to 191}\frac{x^2-(191)^2}{x-(191)}.$$
(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: $\\boxed{382}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=382$. 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 $2a= 382$. |
math-000657 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to -102}\frac{x^2-(-102)^2}{x-(-102)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-204}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-204$. 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 $2a= -204$. |
math-000658 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | 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:
$$\lim_{x\to 159}\frac{x^2-(159)^2}{x-(159)}.$$
(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: $x^2-(159)^2=(x-(159))(x+(159))$.",
"Step 2: For $x\\neq 159$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{318}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=318$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the p... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 318$. |
math-000659 | Calculus: Limits — Removable Discontinuities | 1 | Start by stating any domain restrictions: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 161}\frac{x^2-(161)^2}{x-(161)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a d... | [
{
"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: $\\boxed{322}$.\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 $a$.",
"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 $2a= 322$. (Here the result is $\boxed{322}$.) |
math-000660 | Calculus: Limits — Removable Discontinuities | 1 | Explain what is being counted/optimized: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -322}\frac{x^2-(-322)^2}{x-(-322)}.$$
(a) Evaluate the limit by algebraic simplifica... | [
{
"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: $x^2-(-322)^2=(x-(-322))(x+(-322))$.",
"Step 2: For $x\\neq -322$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. 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 $a$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -644$. (Here the result is $\boxed{-644}$.) |
math-000661 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Exercise: 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:
$$\lim_{x\to 19}\frac{x^2-(19)^2}{x-(19)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative an... | [
{
"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: $\\boxed{38}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=38$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 38$. (Here the result is $\boxed{38}$.) |
math-000662 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to -191}\frac{x^2-(-191)^2}{x-(-191)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limi... | [
{
"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: $\\boxed{-382}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-382$. 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 $2a= -382$. |
math-000663 | Calculus: Limits — Removable Discontinuities | 1 | 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:
$$\lim_{x\to -201}\frac{x^2-(-201)^2}{x-(-201)}.$$
(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: $x^2-(-201)^2=(x-(-201))(x+(-201))$.",
"Step 2: For $x\\neq -201$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-402}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-402$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -402$. (Here the result is $\boxed{-402}$.) |
math-000664 | Calculus: Limits — Removable Discontinuities | 1 | Prompt: 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:
$$\lim_{x\to 119}\frac{x^2-(119)^2}{x-(119)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative a... | [
{
"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: $x^2-(119)^2=(x-(119))(x+(119))$.",
"Step 2: For $x\\neq 119$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{238}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=238$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 238$. |
math-000665 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Warm-up: 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:
$$\lim_{x\to -276}\frac{x^2-(-276)^2}{x-(-276)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-552}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-552$. 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 $2a= -552$. |
math-000666 | Calculus: Limits — Algebraic Simplification | 1 | Show all reasoning: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 149}\frac{x^2-(149)^2}{x-(149)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a ... | [
{
"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: $x^2-(149)^2=(x-(149))(x+(149))$.",
"Step 2: For $x\\neq 149$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{298}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=298$. 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 $2a= 298$. (Here the result is $\boxed{298}$.) |
math-000667 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Track units/moduli 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:
$$\lim_{x\to 90}\frac{x^2-(90)^2}{x-(90)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinter... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{180}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=180$. 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 $2a= 180$. |
math-000668 | Calculus: Limits — Algebraic Simplification | 1 | Provide both a computational and a conceptual explanation: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -378}\frac{x^2-(-378)^2}{x-(-378)}.$$
(a) Evaluate the limit by al... | [
{
"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: $\\boxed{-756}$.\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 $a$.",
"robustness_analysis": "Genera... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -756$. |
math-000669 | Calculus: Limits — Algebraic Simplification | 1 | Question: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 175}\frac{x^2-(175)^2}{x-(175)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that wa... | [
{
"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: $\\boxed{350}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=350$. 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 $2a= 350$. (Here the result is $\boxed{350}$.) |
math-000670 | Calculus: Limits — Difference Quotients | 1 | Give a fully justified solution: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 227}\frac{x^2-(227)^2}{x-(227)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(227)^2=(x-(227))(x+(227))$.",
"Step 2: For $x\\neq 227$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{454}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=454$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustne... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 454$. |
math-000671 | Calculus: Limits — Difference Quotients | 1 | State any required conditions first: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -110}\frac{x^2-(-110)^2}{x-(-110)}.$$
(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": "The two methods are consistent and must coincide. 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": "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 $2a= -220$. (Here the result is $\boxed{-220}$.) |
math-000672 | Calculus: Limits — Difference Quotients | 1 | Give a fully justified solution: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -281}\frac{x^2-(-281)^2}{x-(-281)}.$$
(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: $x^2-(-281)^2=(x-(-281))(x+(-281))$.",
"Step 2: For $x\\neq -281$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. 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 $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -562$. (Here the result is $\boxed{-562}$.) |
math-000673 | Calculus: Limits — Algebraic Simplification | 1 | Give reasoning, not just computation: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -263}\frac{x^2-(-263)^2}{x-(-263)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a de... | [
{
"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: $\\boxed{-526}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-526$. 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 $2a= -526$. (Here the result is $\boxed{-526}$.) |
math-000674 | Calculus: Limits — Algebraic Simplification | 1 | Explain each transformation: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 382}\frac{x^2-(382)^2}{x-(382)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{764}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=764$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustne... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 764$. |
math-000675 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Make each step logically reversible (or explain if not): Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -374}\frac{x^2-(-374)^2}{x-(-374)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-748}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-748$. 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 $2a= -748$. (Here the result is $\boxed{-748}$.) |
math-000676 | Calculus: Limits — Removable Discontinuities | 1 | Track units/moduli carefully: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 84}\frac{x^2-(84)^2}{x-(84)}.$$
(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: $\\boxed{168}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=168$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 168$. (Here the result is $\boxed{168}$.) |
math-000677 | Calculus: Limits — Algebraic Simplification | 1 | Track units/moduli carefully: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 233}\frac{x^2-(233)^2}{x-(233)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative an... | [
{
"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: $\\boxed{466}$.\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_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 $2a= 466$. (Here the result is $\boxed{466}$.) |
math-000678 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Solve (and briefly cross-validate): 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:
$$\lim_{x\to -168}\frac{x^2-(-168)^2}{x-(-168)}.$$
(a) Evaluate the limit by algebraic simplification.... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-336}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-336$. 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 $2a= -336$. |
math-000679 | Calculus: Limits — Algebraic Simplification | 1 | Solve with verification: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -104}\frac{x^2-(-104)^2}{x-(-104)}.$$
(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: $x^2-(-104)^2=(x-(-104))(x+(-104))$.",
"Step 2: For $x\\neq -104$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-208}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-208$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Genera... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -208$. (Here the result is $\boxed{-208}$.) |
math-000680 | Calculus: Limits — Removable Discontinuities | 1 | Be explicit about assumptions: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -359}\frac{x^2-(-359)^2}{x-(-359)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-718}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-718$. 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 $2a= -718$. (Here the result is $\boxed{-718}$.) |
math-000681 | Calculus: Limits — Algebraic Simplification | 1 | Track units/moduli carefully: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 79}\frac{x^2-(79)^2}{x-(79)}.$$
(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: $\\boxed{158}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=158$. 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 $2a= 158$. (Here the result is $\boxed{158}$.) |
math-000682 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Track quantifiers carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -266}\frac{x^2-(-266)^2}{x-(-266)}.$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{-532}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-532$. 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 $2a= -532$. |
math-000683 | Calculus: Limits — Difference Quotients | 1 | Track units/moduli carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -260}\frac{x^2-(-260)^2}{x-(-260)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret th... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-260)^2=(x-(-260))(x+(-260))$.",
"Step 2: For $x\\neq -260$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-520}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-520$. 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 $2a= -520$. (Here the result is $\boxed{-520}$.) |
math-000684 | Calculus: Limits — Algebraic Simplification | 1 | 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:
$$\lim_{x\to -208}\frac{x^2-(-208)^2}{x-(-208)}.$$
(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: $x^2-(-208)^2=(x-(-208))(x+(-208))$.",
"Step 2: For $x\\neq -208$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-416}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-416$. 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 $2a= -416$. (Here the result is $\boxed{-416}$.) |
math-000685 | Calculus: Limits — Algebraic Simplification | 1 | Make each step logically reversible (or explain if not): 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:
$$\lim_{x\to 262}\frac{x^2-(262)^2}{x-(262)}.$$
(a) Evaluate the limit by algebra... | [
{
"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: $\\boxed{524}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=524$. 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 $2a= 524$. (Here the result is $\boxed{524}$.) |
math-000686 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Be explicit about assumptions: 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:
$$\lim_{x\to -122}\frac{x^2-(-122)^2}{x-(-122)}.$$
(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: $x^2-(-122)^2=(x-(-122))(x+(-122))$.",
"Step 2: For $x\\neq -122$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-244}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-244$. 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 $2a= -244$. (Here the result is $\boxed{-244}$.) |
math-000687 | Calculus: Limits — Removable Discontinuities | 1 | 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:
$$\lim_{x\to -385}\frac{x^2-(-385)^2}{x-(-385)}.$$
(a) Evaluate the limit by algebraic simpli... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-770}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-770$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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 $2a= -770$. (Here the result is $\boxed{-770}$.) |
math-000688 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Proceed methodically: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -156}\frac{x^2-(-156)^2}{x-(-156)}.$$
(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: $x^2-(-156)^2=(x-(-156))(x+(-156))$.",
"Step 2: For $x\\neq -156$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-312}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-312$. 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 $2a= -312$. |
math-000689 | Calculus: Limits — Difference Quotients | 1 | Exercise: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -314}\frac{x^2-(-314)^2}{x-(-314)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-628}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-628$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -628$. (Here the result is $\boxed{-628}$.) |
math-000690 | Calculus: Limits — Algebraic Simplification | 1 | 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:
$$\lim_{x\to -232}\frac{x^2-(-232)^2}{x-(-232)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a deriv... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-232)^2=(x-(-232))(x+(-232))$.",
"Step 2: For $x\\neq -232$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-464}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-464$. 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 $2a= -464$. (Here the result is $\boxed{-464}$.) |
math-000691 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | 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:
$$\lim_{x\to 179}\frac{x^2-(179)^2}{x-(179)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"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: $\\boxed{358}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=358$. 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 $2a= 358$. |
math-000692 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | 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:
$$\lim_{x\to -64}\frac{x^2-(-64)^2}{x-(-64)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-128}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-128$. 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 $2a= -128$. (Here the result is $\boxed{-128}$.) |
math-000693 | Calculus: Limits — Removable Discontinuities | 1 | Solve and justify each step: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -362}\frac{x^2-(-362)^2}{x-(-362)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-724}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-724$. 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 $2a= -724$. |
math-000694 | Calculus: Limits — Algebraic Simplification | 1 | Derive the result step-by-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:
$$\lim_{x\to -79}\frac{x^2-(-79)^2}{x-(-79)}.$$
(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: $\\boxed{-158}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-158$. 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 $2a= -158$. (Here the result is $\boxed{-158}$.) |
math-000695 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Indicate where a theorem is used: 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:
$$\lim_{x\to -174}\frac{x^2-(-174)^2}{x-(-174)}.$$
(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: $\\boxed{-348}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-348$. 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 $2a= -348$. |
math-000696 | Calculus: Limits — Removable Discontinuities | 1 | 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:
$$\lim_{x\to -376}\frac{x^2-(-376)^2}{x-(-376)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-752}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-752$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_a... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -752$. (Here the result is $\boxed{-752}$.) |
math-000697 | Calculus: Limits — Removable Discontinuities | 1 | Give reasoning, not just computation: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -383}\frac{x^2-(-383)^2}{x-(-383)}.$$
(a) Evaluate the limit by algebraic simplificatio... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-383)^2=(x-(-383))(x+(-383))$.",
"Step 2: For $x\\neq -383$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. 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_analysis": "If the probl... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -766$. |
math-000698 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Do not skip justification steps: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -72}\frac{x^2-(-72)^2}{x-(-72)}.$$
(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: $\\boxed{-144}$.\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": "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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -144$. |
math-000699 | Calculus: Limits — Algebraic Simplification | 1 | Answer using clear logical steps: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 4}\frac{x^2-(4)^2}{x-(4)}.$$
(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: $x^2-(4)^2=(x-(4))(x+(4))$.",
"Step 2: For $x\\neq 4$, cancel to get $\\frac... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{8}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=8$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustnes... | [
{
"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 $2a= 8$. |
math-000700 | Calculus: Limits — Removable Discontinuities | 1 | Determine the requested value: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -246}\frac{x^2-(-246)^2}{x-(-246)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivativ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-492}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-492$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -492$. (Here the result is $\boxed{-492}$.) |
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