id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-002601 | Calculus: Limits — Difference Quotients | 2 | Give a fully justified solution: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -293}\frac{x^2-(-293)^2}{x-(-293)}.$$
(a) Evaluate the limit by algebraic simplificatio... | [
{
"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: $\\boxed{-586}$.\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_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 $2a= -586$. |
math-002602 | Calculus: Limits — Removable Discontinuities | 2 | Provide a rigorous solution: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -298}\frac{x^2-(-298)^2}{x-(-298)}.$$
(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: $x^2-(-298)^2=(x-(-298))(x+(-298))$.",
"Step 2: For $x\\neq -298$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-596}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-596$. 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 $2a= -596$. |
math-002603 | Calculus: Limits — Difference Quotients | 2 | Do not skip justification steps: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -332}\frac{x^2-(-332)^2}{x-(-332)}.$$
(a) Evaluate the limit by algebraic simplificatio... | [
{
"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: $\\boxed{-664}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-664$. 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 $2a= -664$. |
math-002604 | Calculus: Limits — Difference Quotients | 2 | Solve and then verify: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -361}\frac{x^2-(-361)^2}{x-(-361)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterp... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-722}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-722$. 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 $2a= -722$. (Here the result is $\boxed{-722}$.) |
math-002605 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Write the solution set clearly: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -40}\frac{x^2-(-40)^2}{x-(-40)}.$$
(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: $x^2-(-40)^2=(x-(-40))(x+(-40))$.",
"Step 2: For $x\\neq -40$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-80}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-80$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -80$. |
math-002606 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Show all reasoning: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 2}\frac{x^2-(2)^2}{x-(2)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it tha... | [
{
"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: $x^2-(2)^2=(x-(2))(x+(2))$.",
"Step 2: For $x\\neq 2$, cancel to get $\\frac... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{4}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=4$. 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 $2a= 4$. (Here the result is $\boxed{4}$.) |
math-002607 | Calculus: Limits — Difference Quotients | 2 | Solve with verification: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -264}\frac{x^2-(-264)^2}{x-(-264)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-528}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-528$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensit... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -528$. |
math-002608 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | State any required conditions first: 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:
$$\lim_{x\to -21}\frac{x^2-(-21)^2}{x-(-21)}.$$
(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: $x^2-(-21)^2=(x-(-21))(x+(-21))$.",
"Step 2: For $x\\neq -21$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-42}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-42$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -42$. |
math-002609 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Problem: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 43}\frac{x^2-(43)^2}{x-(43)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit a... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{86}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=86$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustn... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 86$. |
math-002610 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Question: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 106}\frac{x^2-(106)^2}{x-(106)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that wa... | [
{
"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: $x^2-(106)^2=(x-(106))(x+(106))$.",
"Step 2: For $x\\neq 106$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{212}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=212$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 212$. (Here the result is $\boxed{212}$.) |
math-002611 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to 74}\frac{x^2-(74)^2}{x-(74)}.$$
(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: $x^2-(74)^2=(x-(74))(x+(74))$.",
"Step 2: For $x\\neq 74$, cancel to get $\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{148}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=148$. 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 $2a= 148$. |
math-002612 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to 122}\frac{x^2-(122)^2}{x-(122)}.$$
(a) Evaluate the limit by algebraic simplifi... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. 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": "Sensitivity an... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 244$. (Here the result is $\boxed{244}$.) |
math-002613 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Write the solution set clearly: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -32}\frac{x^2-(-32)^2}{x-(-32)}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-64}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-64$. 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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -64$. (Here the result is $\boxed{-64}$.) |
math-002614 | Calculus: Limits — Removable Discontinuities | 2 | Checkpoint: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 285}\frac{x^2-(285)^2}{x-(285)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{570}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=570$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 570$. (Here the result is $\boxed{570}$.) |
math-002615 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to 330}\frac{x^2-(330)^2}{x-(330)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{660}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=660$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 660$. (Here the result is $\boxed{660}$.) |
math-002616 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to -282}\frac{x^2-(-282)^2}{x-(-282)}.$$
(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: $\\boxed{-564}$.\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_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 $2a= -564$. (Here the result is $\boxed{-564}$.) |
math-002617 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Determine the requested 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:
$$\lim_{x\to 189}\frac{x^2-(189)^2}{x-(189)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{378}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=378$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 378$. (Here the result is $\boxed{378}$.) |
math-002618 | Calculus: Limits — Difference Quotients | 2 | Answer with a short justification: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 80}\frac{x^2-(80)^2}{x-(80)}.$$
(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: $x^2-(80)^2=(x-(80))(x+(80))$.",
"Step 2: For $x\\neq 80$, cancel to get $\\... | {
"consistency_check": "Both approaches agree after simplification. 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 $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 160$. (Here the result is $\boxed{160}$.) |
math-002619 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to -14}\frac{x^2-(-14)^2}{x-(-14)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative an... | [
{
"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: $\\boxed{-28}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-28$. 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... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -28$. |
math-002620 | Calculus: Limits — Removable Discontinuities | 2 | Indicate where a theorem is used: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -217}\frac{x^2-(-217)^2}{x-(-217)}.$$
(a) Evaluate the limit by algebraic simplificati... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-434}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-434$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -434$. |
math-002621 | Calculus: Limits — Algebraic Simplification | 2 | 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:
$$\lim_{x\to -133}\frac{x^2-(-133)^2}{x-(-133)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-266}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-266$. 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 $2a= -266$. |
math-002622 | Calculus: Limits — Algebraic Simplification | 2 | Do not skip justification steps: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 306}\frac{x^2-(306)^2}{x-(306)}.$$
(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: $x^2-(306)^2=(x-(306))(x+(306))$.",
"Step 2: For $x\\neq 306$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{612}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=612$. 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 $2a= 612$. (Here the result is $\boxed{612}$.) |
math-002623 | Calculus: Limits — Algebraic Simplification | 2 | Explain what is being counted/optimized: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 362}\frac{x^2-(362)^2}{x-(362)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a de... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(362)^2=(x-(362))(x+(362))$.",
"Step 2: For $x\\neq 362$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. 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 $a$.",
"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 $2a= 724$. (Here the result is $\boxed{724}$.) |
math-002624 | Calculus: Limits — Removable Discontinuities | 2 | 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:
$$\lim_{x\to 357}\frac{x^2-(357)^2}{x-(357)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpre... | [
{
"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 get... | {
"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": "Sensitivity an... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 714$. (Here the result is $\boxed{714}$.) |
math-002625 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Question: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 132}\frac{x^2-(132)^2}{x-(132)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the lim... | [
{
"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: $x^2-(132)^2=(x-(132))(x+(132))$.",
"Step 2: For $x\\neq 132$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{264}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=264$. 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 $2a= 264$. |
math-002626 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Be explicit about assumptions: 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:
$$\lim_{x\to -96}\frac{x^2-(-96)^2}{x-(-96)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-192}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-192$. 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 $2a= -192$. (Here the result is $\boxed{-192}$.) |
math-002627 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Answer using clear logical steps: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 335}\frac{x^2-(335)^2}{x-(335)}.$$
(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: $x^2-(335)^2=(x-(335))(x+(335))$.",
"Step 2: For $x\\neq 335$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{670}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=670$. 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 $2a= 670$. (Here the result is $\boxed{670}$.) |
math-002628 | Calculus: Limits — Difference Quotients | 2 | State any required conditions first: 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:
$$\lim_{x\to -195}\frac{x^2-(-195)^2}{x-(-195)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-390}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-390$. 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 $2a= -390$. |
math-002629 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Do not skip justification steps: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 200}\frac{x^2-(200)^2}{x-(200)}.$$
(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: $\\boxed{400}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=400$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 400$. |
math-002630 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to 62}\frac{x^2-(62)^2}{x-(62)}.$$
(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: $x^2-(62)^2=(x-(62))(x+(62))$.",
"Step 2: For $x\\neq 62$, cancel to get $\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{124}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=124$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 124$. (Here the result is $\boxed{124}$.) |
math-002631 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Give a theorem-based solution: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -209}\frac{x^2-(-209)^2}{x-(-209)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret t... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-418}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-418$. 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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -418$. |
math-002632 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to -352}\frac{x^2-(-352)^2}{x-(-352)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-704}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-704$. 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 $2a= -704$. (Here the result is $\boxed{-704}$.) |
math-002633 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Explain why your operations are valid: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -261}\frac{x^2-(-261)^2}{x-(-261)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a d... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-261)^2=(x-(-261))(x+(-261))$.",
"Step 2: For $x\\neq -261$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-522}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-522$. 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 $2a= -522$. (Here the result is $\boxed{-522}$.) |
math-002634 | Calculus: Limits — Algebraic Simplification | 2 | Prompt: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 81}\frac{x^2-(81)^2}{x-(81)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{162}$.\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 $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 162$. |
math-002635 | Calculus: Limits — Difference Quotients | 2 | Question: 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:
$$\lim_{x\to 236}\frac{x^2-(236)^2}{x-(236)}.$$
(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: $x^2-(236)^2=(x-(236))(x+(236))$.",
"Step 2: For $x\\neq 236$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{472}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=472$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 472$. |
math-002636 | Calculus: Limits — Algebraic Simplification | 2 | Problem: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 75}\frac{x^2-(75)^2}{x-(75)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit 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: $x^2-(75)^2=(x-(75))(x+(75))$.",
"Step 2: For $x\\neq 75$, cancel to get $\\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{150}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=150$. 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 $2a= 150$. (Here the result is $\boxed{150}$.) |
math-002637 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to 223}\frac{x^2-(223)^2}{x-(223)}.$$
(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: $\\boxed{446}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=446$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= 446$. (Here the result is $\boxed{446}$.) |
math-002638 | Calculus: Limits — Removable Discontinuities | 2 | Solve and then verify: 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:
$$\lim_{x\to 86}\frac{x^2-(86)^2}{x-(86)}.$$
(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: $\\boxed{172}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=172$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the p... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 172$. |
math-002639 | Calculus: Limits — Difference Quotients | 2 | Answer with a short justification: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 343}\frac{x^2-(343)^2}{x-(343)}.$$
(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: $x^2-(343)^2=(x-(343))(x+(343))$.",
"Step 2: For $x\\neq 343$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{686}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=686$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 686$. |
math-002640 | Calculus: Limits — Algebraic Simplification | 2 | Indicate where a theorem is used: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -52}\frac{x^2-(-52)^2}{x-(-52)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret t... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-104}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-104$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -104$. |
math-002641 | Calculus: Limits — Removable Discontinuities | 2 | Start by stating any domain restrictions: 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:
$$\lim_{x\to -173}\frac{x^2-(-173)^2}{x-(-173)}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-346}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-346$. 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 $2a= -346$. (Here the result is $\boxed{-346}$.) |
math-002642 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Compute the requested quantity: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -22}\frac{x^2-(-22)^2}{x-(-22)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-44}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-44$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -44$. (Here the result is $\boxed{-44}$.) |
math-002643 | Calculus: Limits — Difference Quotients | 2 | Give an answer and a quick 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:
$$\lim_{x\to -311}\frac{x^2-(-311)^2}{x-(-311)}.$$
(a) Evaluate the limit by algebraic simplifica... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-622}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-622$. 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 $2a= -622$. (Here the result is $\boxed{-622}$.) |
math-002644 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to 139}\frac{x^2-(139)^2}{x-(139)}.$$
(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: $x^2-(139)^2=(x-(139))(x+(139))$.",
"Step 2: For $x\\neq 139$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{278}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=278$. 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 $2a= 278$. (Here the result is $\boxed{278}$.) |
math-002645 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Compute the requested quantity: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 83}\frac{x^2-(83)^2}{x-(83)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reint... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(83)^2=(x-(83))(x+(83))$.",
"Step 2: For $x\\neq 83$, cancel to get $\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{166}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=166$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 166$. (Here the result is $\boxed{166}$.) |
math-002646 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Give an answer and a quick 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:
$$\lim_{x\to 196}\frac{x^2-(196)^2}{x-(196)}.$$
(a) Evaluate the limit by algebraic simplificatio... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{392}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=392$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 392$. |
math-002647 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Provide both a computational and a conceptual explanation: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -335}\frac{x^2-(-335)^2}{x-(-335)}.$$
(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: $x^2-(-335)^2=(x-(-335))(x+(-335))$.",
"Step 2: For $x\\neq -335$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-670}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-670$. 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 $2a= -670$. (Here the result is $\boxed{-670}$.) |
math-002648 | Calculus: Limits — Algebraic Simplification | 2 | Try to avoid pattern-matching; explain why: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 58}\frac{x^2-(58)^2}{x-(58)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{116}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=116$. 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 $2a= 116$. (Here the result is $\boxed{116}$.) |
math-002649 | Calculus: Limits — Algebraic Simplification | 2 | Solve and include a self-check: 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:
$$\lim_{x\to -44}\frac{x^2-(-44)^2}{x-(-44)}.$$
(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: $\\boxed{-88}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-88$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitivity an... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -88$. (Here the result is $\boxed{-88}$.) |
math-002650 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Challenge: 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 a... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{110}$.\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 $a$.",
"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 $2a= 110$. (Here the result is $\boxed{110}$.) |
math-002651 | Calculus: Limits — Algebraic Simplification | 2 | Start by stating any domain restrictions: 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:
$$\lim_{x\to 157}\frac{x^2-(157)^2}{x-(157)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reint... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{314}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=314$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 314$. (Here the result is $\boxed{314}$.) |
math-002652 | Calculus: Limits — Removable Discontinuities | 2 | Solve and then verify: 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:
$$\lim_{x\to -364}\frac{x^2-(-364)^2}{x-(-364)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-364)^2=(x-(-364))(x+(-364))$.",
"Step 2: For $x\\neq -364$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-728}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-728$. 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 $2a= -728$. (Here the result is $\boxed{-728}$.) |
math-002653 | Calculus: Limits — Difference Quotients | 2 | Make each step logically reversible (or explain if not): Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 99}\frac{x^2-(99)^2}{x-(99)}.$$
(a) Evaluate the limit by algeb... | [
{
"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: $\\boxed{198}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=198$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the 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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 198$. (Here the result is $\boxed{198}$.) |
math-002654 | Calculus: Limits — Removable Discontinuities | 2 | Track quantifiers carefully: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -121}\frac{x^2-(-121)^2}{x-(-121)}.$$
(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: $x^2-(-121)^2=(x-(-121))(x+(-121))$.",
"Step 2: For $x\\neq -121$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-242}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-242$. 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 $2a= -242$. (Here the result is $\boxed{-242}$.) |
math-002655 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Provide both a computational and a conceptual explanation: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 286}\frac{x^2-(286)^2}{x-(286)}.$$
(a) Evaluate the limit by ... | [
{
"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: $\\boxed{572}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=572$. 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 $2a= 572$. (Here the result is $\boxed{572}$.) |
math-002656 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Carefully track domains: 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:
$$\lim_{x\to -396}\frac{x^2-(-396)^2}{x-(-396)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the lim... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-792}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-792$. 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 $2a= -792$. (Here the result is $\boxed{-792}$.) |
math-002657 | Calculus: Limits — Algebraic Simplification | 2 | Answer with a short justification: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 202}\frac{x^2-(202)^2}{x-(202)}.$$
(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: $\\boxed{404}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=404$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 404$. |
math-002658 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Solve and include a self-check: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 252}\frac{x^2-(252)^2}{x-(252)}.$$
(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: $\\boxed{504}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=504$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 504$. (Here the result is $\boxed{504}$.) |
math-002659 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to -24}\frac{x^2-(-24)^2}{x-(-24)}.$$
(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: $x^2-(-24)^2=(x-(-24))(x+(-24))$.",
"Step 2: For $x\\neq -24$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-48}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-48$. 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 $2a= -48$. (Here the result is $\boxed{-48}$.) |
math-002660 | Calculus: Limits — Removable Discontinuities | 2 | Solve and then verify: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -214}\frac{x^2-(-214)^2}{x-(-214)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and co... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-214)^2=(x-(-214))(x+(-214))$.",
"Step 2: For $x\\neq -214$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. 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 $a$.",
"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 $2a= -428$. |
math-002661 | Calculus: Limits — Algebraic Simplification | 2 | Solve and justify each 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:
$$\lim_{x\to 276}\frac{x^2-(276)^2}{x-(276)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{552}$.\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 $a$.",
"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 $2a= 552$. |
math-002662 | Calculus: Limits — Removable Discontinuities | 2 | Provide both a computational and a conceptual explanation: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 114}\frac{x^2-(114)^2}{x-(114)}.$$
(a) Evaluate the limit by algebraic simplifi... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{228}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=228$. 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 $2a= 228$. |
math-002663 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Solve and justify each step: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -200}\frac{x^2-(-200)^2}{x-(-200)}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-400}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-400$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensit... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -400$. |
math-002664 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | 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:
$$\lim_{x\to 146}\frac{x^2-(146)^2}{x-(146)}.$$
(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: $\\boxed{292}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=292$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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 $2a= 292$. |
math-002665 | Calculus: Limits — Algebraic Simplification | 2 | Do not skip justification 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:
$$\lim_{x\to 389}\frac{x^2-(389)^2}{x-(389)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret th... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{778}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=778$. 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 $2a= 778$. (Here the result is $\boxed{778}$.) |
math-002666 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to -267}\frac{x^2-(-267)^2}{x-(-267)}.$$
(a) Evaluate the limit by algebraic simplifica... | [
{
"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: $\\boxed{-534}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-534$. 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 $2a= -534$. (Here the result is $\boxed{-534}$.) |
math-002667 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Solve and justify each step: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 220}\frac{x^2-(220)^2}{x-(220)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{440}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=440$. 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 $2a= 440$. (Here the result is $\boxed{440}$.) |
math-002668 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to 251}\frac{x^2-(251)^2}{x-(251)}.$$
(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: $x^2-(251)^2=(x-(251))(x+(251))$.",
"Step 2: For $x\\neq 251$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{502}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=502$. 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 $2a= 502$. |
math-002669 | Calculus: Limits — Removable Discontinuities | 2 | State any required conditions first: 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:
$$\lim_{x\to -239}\frac{x^2-(-239)^2}{x-(-239)}.$$
(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: $\\boxed{-478}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-478$. 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 $2a= -478$. (Here the result is $\boxed{-478}$.) |
math-002670 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Provide a rigorous solution: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -103}\frac{x^2-(-103)^2}{x-(-103)}.$$
(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: $\\boxed{-206}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-206$. 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 $2a= -206$. |
math-002671 | Calculus: Limits — Difference Quotients | 2 | Question: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -123}\frac{x^2-(-123)^2}{x-(-123)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-246}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-246$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -246$. (Here the result is $\boxed{-246}$.) |
math-002672 | Calculus: Limits — Difference Quotients | 2 | Keep the final answer in boxed form: 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:
$$\lim_{x\to 347}\frac{x^2-(347)^2}{x-(347)}.$$
(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: $x^2-(347)^2=(x-(347))(x+(347))$.",
"Step 2: For $x\\neq 347$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{694}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=694$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the 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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 694$. (Here the result is $\boxed{694}$.) |
math-002673 | Calculus: Limits — Algebraic Simplification | 2 | Solve and then verify: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 44}\frac{x^2-(44)^2}{x-(44)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{88}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=88$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness note:... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 88$. (Here the result is $\boxed{88}$.) |
math-002674 | Calculus: Limits — Removable Discontinuities | 2 | Provide a rigorous solution: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -194}\frac{x^2-(-194)^2}{x-(-194)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-388}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-388$. 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 $2a= -388$. |
math-002675 | Calculus: Limits — Algebraic Simplification | 2 | Work carefully and justify each inference: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 373}\frac{x^2-(373)^2}{x-(373)}.$$
(a) Evaluate the limit by algebraic simplificat... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. 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 $a$.",
"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 $2a= 746$. (Here the result is $\boxed{746}$.) |
math-002676 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Start by stating any domain restrictions: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -126}\frac{x^2-(-126)^2}{x-(-126)}.$$
(a) Evaluate the limit by algebraic simp... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-252}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-252$. 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... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -252$. |
math-002677 | Calculus: Limits — Algebraic Simplification | 2 | 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:
$$\lim_{x\to -15}\frac{x^2-(-15)^2}{x-(-15)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-30}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-30$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -30$. (Here the result is $\boxed{-30}$.) |
math-002678 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Solve (and briefly cross-validate): Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -90}\frac{x^2-(-90)^2}{x-(-90)}.$$
(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: $\\boxed{-180}$.\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 $a$.",
"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 $2a= -180$. (Here the result is $\boxed{-180}$.) |
math-002679 | Calculus: Limits — Removable Discontinuities | 2 | Find the exact value: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -30}\frac{x^2-(-30)^2}{x-(-30)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-30)^2=(x-(-30))(x+(-30))$.",
"Step 2: For $x\\neq -30$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-60}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-60$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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 $2a= -60$. |
math-002680 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Proceed methodically: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 247}\frac{x^2-(247)^2}{x-(247)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{494}$.\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_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... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 494$. (Here the result is $\boxed{494}$.) |
math-002681 | Calculus: Limits — Removable Discontinuities | 2 | Carefully track domains: 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:
$$\lim_{x\to 14}\frac{x^2-(14)^2}{x-(14)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{28}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=28$. 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 $2a= 28$. (Here the result is $\boxed{28}$.) |
math-002682 | Calculus: Limits — Algebraic Simplification | 2 | Compute the requested quantity: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 337}\frac{x^2-(337)^2}{x-(337)}.$$
(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: $x^2-(337)^2=(x-(337))(x+(337))$.",
"Step 2: For $x\\neq 337$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{674}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=674$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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 $2a= 674$. (Here the result is $\boxed{674}$.) |
math-002683 | Calculus: Limits — Removable Discontinuities | 2 | Answer using clear logical 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:
$$\lim_{x\to -131}\frac{x^2-(-131)^2}{x-(-131)}.$$
(a) Evaluate the limit by algebraic simplification.
(... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-262}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-262$. 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 $2a= -262$. (Here the result is $\boxed{-262}$.) |
math-002684 | Calculus: Limits — Difference Quotients | 2 | Be explicit about assumptions: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 66}\frac{x^2-(66)^2}{x-(66)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{132}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=132$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= 132$. |
math-002685 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to -258}\frac{x^2-(-258)^2}{x-(-258)}.$$
(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: $\\boxed{-516}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-516$. 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... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -516$. |
math-002686 | Calculus: Limits — Algebraic Simplification | 2 | Derive the result step-by-step: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -369}\frac{x^2-(-369)^2}{x-(-369)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivati... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-738}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-738$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensit... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -738$. (Here the result is $\boxed{-738}$.) |
math-002687 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | Solve and sanity-check: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -51}\frac{x^2-(-51)^2}{x-(-51)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reint... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-102}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-102$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality 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 $2a= -102$. |
math-002688 | Calculus: Limits — Removable Discontinuities | 2 | 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:
$$\lim_{x\to 210}\frac{x^2-(210)^2}{x-(210)}.$$
(a) Evaluate the limit by algebraic simplific... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{420}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=420$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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 $2a= 420$. (Here the result is $\boxed{420}$.) |
math-002689 | Calculus: Limits — Algebraic Simplification | 2 | 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:
$$\lim_{x\to 73}\frac{x^2-(73)^2}{x-(73)}.$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{146}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=146$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 146$. (Here the result is $\boxed{146}$.) |
math-002690 | Calculus: Limits — Secant-to-Tangent Interpretation | 2 | 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:
$$\lim_{x\to -351}\frac{x^2-(-351)^2}{x-(-351)}.$$
(a) Evaluate the limit by algebraic simplification.
(b... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-702}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-702$. 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 $2a= -702$. (Here the result is $\boxed{-702}$.) |
math-002691 | Calculus: Limits — Removable Discontinuities | 2 | Solve and justify each step: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 356}\frac{x^2-(356)^2}{x-(356)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{712}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=712$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 712$. (Here the result is $\boxed{712}$.) |
math-002692 | Calculus: Limits — Removable Discontinuities | 2 | Warm-up: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 387}\frac{x^2-(387)^2}{x-(387)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limi... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(387)^2=(x-(387))(x+(387))$.",
"Step 2: For $x\\neq 387$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{774}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=774$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 774$. (Here the result is $\boxed{774}$.) |
math-002693 | Calculus: Limits — Algebraic Simplification | 2 | Be explicit about assumptions: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 131}\frac{x^2-(131)^2}{x-(131)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(131)^2=(x-(131))(x+(131))$.",
"Step 2: For $x\\neq 131$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{262}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=262$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 262$. |
math-002694 | Calculus: Limits — Algebraic Simplification | 2 | Compute the requested quantity: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 284}\frac{x^2-(284)^2}{x-(284)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{568}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=568$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 568$. (Here the result is $\boxed{568}$.) |
math-002695 | Calculus: Limits — Difference Quotients | 2 | Question: 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:
$$\lim_{x\to -26}\frac{x^2-(-26)^2}{x-(-26)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-26)^2=(x-(-26))(x+(-26))$.",
"Step 2: For $x\\neq -26$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-52}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-52$. 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 $2a= -52$. |
math-002696 | Calculus: Limits — Algebraic Simplification | 2 | State any required conditions first: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 274}\frac{x^2-(274)^2}{x-(274)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a deriva... | [
{
"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: $\\boxed{548}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=548$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robus... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 548$. (Here the result is $\boxed{548}$.) |
math-002697 | Calculus: Limits — Difference Quotients | 2 | 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:
$$\lim_{x\to -29}\frac{x^2-(-29)^2}{x-(-29)}.$$
(a) Evaluate the limit by algebra... | [
{
"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: $x^2-(-29)^2=(x-(-29))(x+(-29))$.",
"Step 2: For $x\\neq -29$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-58}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-58$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -58$. (Here the result is $\boxed{-58}$.) |
math-002698 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Exercise: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -63}\frac{x^2-(-63)^2}{x-(-63)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that wa... | [
{
"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: $x^2-(-63)^2=(x-(-63))(x+(-63))$.",
"Step 2: For $x\\neq -63$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-126}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-126$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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 $2a= -126$. |
math-002699 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Where appropriate, name the theorem you use: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 89}\frac{x^2-(89)^2}{x-(89)}.$$
(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: $\\boxed{178}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=178$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the p... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 178$. (Here the result is $\boxed{178}$.) |
math-002700 | Calculus: Limits — Indeterminate Forms (0/0) | 2 | Keep the final answer in boxed form: 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:
$$\lim_{x\to -329}\frac{x^2-(-329)^2}{x-(-329)}.$$
(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: $\\boxed{-658}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-658$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensit... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -658$. (Here the result is $\boxed{-658}$.) |
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