id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-000701 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Give an answer and a quick verification: 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:
$$\lim_{x\to -372}\frac{x^2-(-372)^2}{x-(-372)}.$$
(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: $x^2-(-372)^2=(x-(-372))(x+(-372))$.",
"Step 2: For $x\\neq -372$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-744}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-744$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -744$. |
math-000702 | Calculus: Limits — Algebraic Simplification | 1 | Where appropriate, name the theorem you use: 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:
$$\lim_{x\to 313}\frac{x^2-(313)^2}{x-(313)}.$$
(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: $x^2-(313)^2=(x-(313))(x+(313))$.",
"Step 2: For $x\\neq 313$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{626}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=626$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= 626$. (Here the result is $\boxed{626}$.) |
math-000703 | Calculus: Limits — Algebraic Simplification | 1 | Give a fully justified solution: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -303}\frac{x^2-(-303)^2}{x-(-303)}.$$
(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: $\\boxed{-606}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-606$. 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 $2a= -606$. (Here the result is $\boxed{-606}$.) |
math-000704 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | 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:
$$\lim_{x\to -310}\frac{x^2-(-310)^2}{x-(-310)}.$$
(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: $\\boxed{-620}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-620$. 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 $2a= -620$. |
math-000705 | Calculus: Limits — Difference Quotients | 1 | Carefully track domains: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -202}\frac{x^2-(-202)^2}{x-(-202)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinte... | [
{
"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: $\\boxed{-404}$.\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": "Robust... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -404$. (Here the result is $\boxed{-404}$.) |
math-000706 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Checkpoint: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -345}\frac{x^2-(-345)^2}{x-(-345)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-690}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-690$. 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 $2a= -690$. |
math-000707 | Calculus: Limits — Algebraic Simplification | 1 | Provide a rigorous solution: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 40}\frac{x^2-(40)^2}{x-(40)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterp... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{80}$.\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 $a$.",
"robustness_analysis": "Sensitivit... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 80$. (Here the result is $\boxed{80}$.) |
math-000708 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to 20}\frac{x^2-(20)^2}{x-(20)}.$$
(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: $x^2-(20)^2=(x-(20))(x+(20))$.",
"Step 2: For $x\\neq 20$, cancel to get $\\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{40}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=40$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the problem w... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 40$. |
math-000709 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to 224}\frac{x^2-(224)^2}{x-(224)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{448}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=448$. 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 $2a= 448$. |
math-000710 | Calculus: Limits — Algebraic Simplification | 1 | Question: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 324}\frac{x^2-(324)^2}{x-(324)}.$$
(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: $x^2-(324)^2=(x-(324))(x+(324))$.",
"Step 2: For $x\\neq 324$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{648}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=648$. 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 $2a= 648$. (Here the result is $\boxed{648}$.) |
math-000711 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Give an answer and a quick verification: 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) 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: $x^2-(396)^2=(x-(396))(x+(396))$.",
"Step 2: For $x\\neq 396$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. 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_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 $2a= 792$. |
math-000712 | Calculus: Limits — Difference Quotients | 1 | Show all reasoning: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 317}\frac{x^2-(317)^2}{x-(317)}.$$
(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: $\\boxed{634}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=634$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 634$. (Here the result is $\boxed{634}$.) |
math-000713 | Calculus: Limits — Difference Quotients | 1 | Track quantifiers 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:
$$\lim_{x\to -127}\frac{x^2-(-127)^2}{x-(-127)}.$$
(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: $x^2-(-127)^2=(x-(-127))(x+(-127))$.",
"Step 2: For $x\\neq -127$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-254}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-254$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -254$. |
math-000714 | Calculus: Limits — Difference Quotients | 1 | Provide a rigorous solution: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 392}\frac{x^2-(392)^2}{x-(392)}.$$
(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: $x^2-(392)^2=(x-(392))(x+(392))$.",
"Step 2: For $x\\neq 392$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{784}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=784$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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 $2a= 784$. (Here the result is $\boxed{784}$.) |
math-000715 | Calculus: Limits — Removable Discontinuities | 1 | Determine the requested value: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 166}\frac{x^2-(166)^2}{x-(166)}.$$
(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: $x^2-(166)^2=(x-(166))(x+(166))$.",
"Step 2: For $x\\neq 166$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{332}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=332$. 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 $2a= 332$. |
math-000716 | Calculus: Limits — Difference Quotients | 1 | Do not skip justification steps: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 127}\frac{x^2-(127)^2}{x-(127)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{254}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=254$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality not... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 254$. (Here the result is $\boxed{254}$.) |
math-000717 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Task: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -53}\frac{x^2-(-53)^2}{x-(-53)}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{-106}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-106$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensit... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -106$. |
math-000718 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | 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:
$$\lim_{x\to -270}\frac{x^2-(-270)^2}{x-(-270)}.$$
(a) Evaluate the limit by... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-540}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-540$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -540$. (Here the result is $\boxed{-540}$.) |
math-000719 | Calculus: Limits — Algebraic Simplification | 1 | Checkpoint: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 377}\frac{x^2-(377)^2}{x-(377)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{754}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=754$. 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 $2a= 754$. |
math-000720 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Provide a rigorous solution: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 34}\frac{x^2-(34)^2}{x-(34)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterp... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{68}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=68$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analy... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 68$. (Here the result is $\boxed{68}$.) |
math-000721 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Checkpoint: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 212}\frac{x^2-(212)^2}{x-(212)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{424}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=424$. 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 $2a= 424$. |
math-000722 | Calculus: Limits — Difference Quotients | 1 | Track units/moduli carefully: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 319}\frac{x^2-(319)^2}{x-(319)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rein... | [
{
"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: $x^2-(319)^2=(x-(319))(x+(319))$.",
"Step 2: For $x\\neq 319$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{638}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=638$. 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 $2a= 638$. |
math-000723 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Determine the requested value: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 249}\frac{x^2-(249)^2}{x-(249)}.$$
(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: $x^2-(249)^2=(x-(249))(x+(249))$.",
"Step 2: For $x\\neq 249$, cancel to get... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{498}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=498$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 498$. |
math-000724 | Calculus: Limits — Removable Discontinuities | 1 | Warm-up: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -253}\frac{x^2-(-253)^2}{x-(-253)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-506}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-506$. 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 $2a= -506$. |
math-000725 | Calculus: Limits — Removable Discontinuities | 1 | Track units/moduli carefully: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 245}\frac{x^2-(245)^2}{x-(245)}.$$
(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: $\\boxed{490}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=490$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 490$. (Here the result is $\boxed{490}$.) |
math-000726 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Checkpoint: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 115}\frac{x^2-(115)^2}{x-(115)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the l... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{230}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=230$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 230$. (Here the result is $\boxed{230}$.) |
math-000727 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Warm-up: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -147}\frac{x^2-(-147)^2}{x-(-147)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that ... | [
{
"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: $x^2-(-147)^2=(x-(-147))(x+(-147))$.",
"Step 2: For $x\\neq -147$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-294}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-294$. 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 $2a= -294$. |
math-000728 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Task: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -225}\frac{x^2-(-225)^2}{x-(-225)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) 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: $x^2-(-225)^2=(x-(-225))(x+(-225))$.",
"Step 2: For $x\\neq -225$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-450}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-450$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -450$. |
math-000729 | Calculus: Limits — Removable Discontinuities | 1 | 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:
$$\lim_{x\to 399}\frac{x^2-(399)^2}{x-(399)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{798}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=798$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the 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 $2a= 798$. (Here the result is $\boxed{798}$.) |
math-000730 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | State any required conditions first: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 243}\frac{x^2-(243)^2}{x-(243)}.$$
(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: $x^2-(243)^2=(x-(243))(x+(243))$.",
"Step 2: For $x\\neq 243$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{486}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=486$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 486$. (Here the result is $\boxed{486}$.) |
math-000731 | Calculus: Limits — Difference Quotients | 1 | Solve (and briefly cross-validate): 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:
$$\lim_{x\to 316}\frac{x^2-(316)^2}{x-(316)}.$$
(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: $x^2-(316)^2=(x-(316))(x+(316))$.",
"Step 2: For $x\\neq 316$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{632}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=632$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 632$. (Here the result is $\boxed{632}$.) |
math-000732 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to -354}\frac{x^2-(-354)^2}{x-(-354)}.$$
(a) Evaluate the limit by algebraic simplif... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-708}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-708$. 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 $2a= -708$. (Here the result is $\boxed{-708}$.) |
math-000733 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Warm-up: 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:
$$\lim_{x\to -285}\frac{x^2-(-285)^2}{x-(-285)}.$$
(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": "Consistency verification shows both paths yield the identical boxed result. 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 $a$.",
"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 $2a= -570$. |
math-000734 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Be explicit about assumptions: 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:
$$\lim_{x\to 230}\frac{x^2-(230)^2}{x-(230)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{460}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=460$. 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 $2a= 460$. |
math-000735 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Derive the result step-by-step: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 173}\frac{x^2-(173)^2}{x-(173)}.$$
(a) Evaluate the limit by algebraic simplification.
(... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{346}$.\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 $a$.",
"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 $2a= 346$. (Here the result is $\boxed{346}$.) |
math-000736 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Task: 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:
$$\lim_{x\to -216}\frac{x^2-(-216)^2}{x-(-216)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limi... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-432}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-432$. 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 $2a= -432$. (Here the result is $\boxed{-432}$.) |
math-000737 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Warm-up: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -137}\frac{x^2-(-137)^2}{x-(-137)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that ... | [
{
"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: $x^2-(-137)^2=(x-(-137))(x+(-137))$.",
"Step 2: For $x\\neq -137$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-274}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-274$. 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 $2a= -274$. (Here the result is $\boxed{-274}$.) |
math-000738 | Calculus: Limits — Algebraic Simplification | 1 | Track quantifiers carefully: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 13}\frac{x^2-(13)^2}{x-(13)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterp... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{26}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=26$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analy... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 26$. |
math-000739 | Calculus: Limits — Removable Discontinuities | 1 | Compute the requested quantity: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -179}\frac{x^2-(-179)^2}{x-(-179)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-358}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-358$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -358$. (Here the result is $\boxed{-358}$.) |
math-000740 | Calculus: Limits — Removable Discontinuities | 1 | Be explicit about assumptions: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 134}\frac{x^2-(134)^2}{x-(134)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{268}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=268$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 268$. (Here the result is $\boxed{268}$.) |
math-000741 | Calculus: Limits — Algebraic Simplification | 1 | Make each step logically reversible (or explain if not): Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -101}\frac{x^2-(-101)^2}{x-(-101)}.$$
(a) Evaluate the limit by alge... | [
{
"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: $\\boxed{-202}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-202$. 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 $2a= -202$. |
math-000742 | Calculus: Limits — Difference Quotients | 1 | Work this out carefully: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -61}\frac{x^2-(-61)^2}{x-(-61)}.$$
(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: $\\boxed{-122}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-122$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -122$. |
math-000743 | Calculus: Limits — Difference Quotients | 1 | Give a theorem-based solution: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -27}\frac{x^2-(-27)^2}{x-(-27)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative a... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-54}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-54$. 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 $2a= -54$. (Here the result is $\boxed{-54}$.) |
math-000744 | Calculus: Limits — Removable Discontinuities | 1 | Explain what is being counted/optimized: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -348}\frac{x^2-(-348)^2}{x-(-348)}.$$
(a) Evaluate the limit by algebraic simplifica... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-348)^2=(x-(-348))(x+(-348))$.",
"Step 2: For $x\\neq -348$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-696}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-696$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -696$. (Here the result is $\boxed{-696}$.) |
math-000745 | Calculus: Limits — Removable Discontinuities | 1 | Provide both a computational and a conceptual explanation: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -324}\frac{x^2-(-324)^2}{x-(-324)}.$$
(a) Evaluate the limit by algebraic simpl... | [
{
"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: $\\boxed{-648}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-648$. 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 $2a= -648$. (Here the result is $\boxed{-648}$.) |
math-000746 | Calculus: Limits — Removable Discontinuities | 1 | 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:
$$\lim_{x\to 23}\frac{x^2-(23)^2}{x-(23)}.$$
(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: $x^2-(23)^2=(x-(23))(x+(23))$.",
"Step 2: For $x\\neq 23$, cancel to get $\\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{46}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=46$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analy... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 46$. (Here the result is $\boxed{46}$.) |
math-000747 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | 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:
$$\lim_{x\to 180}\frac{x^2-(180)^2}{x-(180)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{360}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=360$. 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 $2a= 360$. (Here the result is $\boxed{360}$.) |
math-000748 | Calculus: Limits — Algebraic Simplification | 1 | Use two approaches if possible: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -74}\frac{x^2-(-74)^2}{x-(-74)}.$$
(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 $a$.",
"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 $2a= -148$. (Here the result is $\boxed{-148}$.) |
math-000749 | Calculus: Limits — Removable Discontinuities | 1 | Problem: 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:
$$\lim_{x\to -278}\frac{x^2-(-278)^2}{x-(-278)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivati... | [
{
"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: $x^2-(-278)^2=(x-(-278))(x+(-278))$.",
"Step 2: For $x\\neq -278$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-556}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-556$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -556$. (Here the result is $\boxed{-556}$.) |
math-000750 | Calculus: Limits — Difference Quotients | 1 | Proceed methodically: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -135}\frac{x^2-(-135)^2}{x-(-135)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and com... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-270}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-270$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -270$. |
math-000751 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Track quantifiers 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:
$$\lim_{x\to -132}\frac{x^2-(-132)^2}{x-(-132)}.$$
(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: $\\boxed{-264}$.\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_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 $2a= -264$. |
math-000752 | Calculus: Limits — Difference Quotients | 1 | Checkpoint: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -111}\frac{x^2-(-111)^2}{x-(-111)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-222}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-222$. 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 $2a= -222$. (Here the result is $\boxed{-222}$.) |
math-000753 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Answer using clear logical steps: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -400}\frac{x^2-(-400)^2}{x-(-400)}.$$
(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: $\\boxed{-800}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-800$. 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 $2a= -800$. (Here the result is $\boxed{-800}$.) |
math-000754 | Calculus: Limits — Removable Discontinuities | 1 | Determine the requested value: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 265}\frac{x^2-(265)^2}{x-(265)}.$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{530}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=530$. 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 $2a= 530$. (Here the result is $\boxed{530}$.) |
math-000755 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Proceed methodically: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 257}\frac{x^2-(257)^2}{x-(257)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{514}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=514$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= 514$. (Here the result is $\boxed{514}$.) |
math-000756 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to 323}\frac{x^2-(323)^2}{x-(323)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{646}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=646$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Sensitiv... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 646$. (Here the result is $\boxed{646}$.) |
math-000757 | Calculus: Limits — Algebraic Simplification | 1 | 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:
$$\lim_{x\to -379}\frac{x^2-(-379)^2}{x-(-379)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compu... | [
{
"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: $\\boxed{-758}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-758$. 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 $2a= -758$. (Here the result is $\boxed{-758}$.) |
math-000758 | Calculus: Limits — Algebraic Simplification | 1 | Track quantifiers carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -39}\frac{x^2-(-39)^2}{x-(-39)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the li... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-78}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-78$. 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 $2a= -78$. |
math-000759 | Calculus: Limits — Difference Quotients | 1 | Answer using clear logical steps: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 111}\frac{x^2-(111)^2}{x-(111)}.$$
(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: $x^2-(111)^2=(x-(111))(x+(111))$.",
"Step 2: For $x\\neq 111$, cancel to get... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{222}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=222$. 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 $2a= 222$. (Here the result is $\boxed{222}$.) |
math-000760 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | 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:
$$\lim_{x\to 5}\frac{x^2-(5)^2}{x-(5)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{10}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=10$. 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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 10$. |
math-000761 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Indicate where a theorem is used: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 277}\frac{x^2-(277)^2}{x-(277)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{554}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=554$. 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 $2a= 554$. (Here the result is $\boxed{554}$.) |
math-000762 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | 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:
$$\lim_{x\to 386}\frac{x^2-(386)^2}{x-(386)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{772}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=772$. 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 $2a= 772$. |
math-000763 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Indicate where a theorem is used: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -215}\frac{x^2-(-215)^2}{x-(-215)}.$$
(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: $\\boxed{-430}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-430$. 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 $2a= -430$. |
math-000764 | Calculus: Limits — Difference Quotients | 1 | Derive the result step-by-step: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 222}\frac{x^2-(222)^2}{x-(222)}.$$
(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: $x^2-(222)^2=(x-(222))(x+(222))$.",
"Step 2: For $x\\neq 222$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{444}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=444$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 444$. |
math-000765 | Calculus: Limits — Algebraic Simplification | 1 | Solve and sanity-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:
$$\lim_{x\to 169}\frac{x^2-(169)^2}{x-(169)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpre... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{338}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=338$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= 338$. (Here the result is $\boxed{338}$.) |
math-000766 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Track quantifiers carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 102}\frac{x^2-(102)^2}{x-(102)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the li... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{204}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=204$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 204$. (Here the result is $\boxed{204}$.) |
math-000767 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Answer using clear logical steps: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -296}\frac{x^2-(-296)^2}{x-(-296)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpre... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-592}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-592$. 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 $2a= -592$. (Here the result is $\boxed{-592}$.) |
math-000768 | Calculus: Limits — Algebraic Simplification | 1 | Checkpoint: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 168}\frac{x^2-(168)^2}{x-(168)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{336}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=336$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the 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 $2a= 336$. (Here the result is $\boxed{336}$.) |
math-000769 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Solve and sanity-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:
$$\lim_{x\to 358}\frac{x^2-(358)^2}{x-(358)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpre... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{716}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=716$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= 716$. (Here the result is $\boxed{716}$.) |
math-000770 | Calculus: Limits — Removable Discontinuities | 1 | Exercise: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 271}\frac{x^2-(271)^2}{x-(271)}.$$
(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: $x^2-(271)^2=(x-(271))(x+(271))$.",
"Step 2: For $x\\neq 271$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{542}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=542$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "If the p... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Key idea: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 542$. (Here the result is $\boxed{542}$.) |
math-000771 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | 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:
$$\lim_{x\to -220}\frac{x^2-(-220)^2}{x-(-220)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compu... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-220)^2=(x-(-220))(x+(-220))$.",
"Step 2: For $x\\neq -220$, cancel to... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. 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 $a$.",
"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 $2a= -440$. (Here the result is $\boxed{-440}$.) |
math-000772 | Calculus: Limits — Algebraic Simplification | 1 | Challenge: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -98}\frac{x^2-(-98)^2}{x-(-98)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit a... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-196}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-196$. 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 $2a= -196$. |
math-000773 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to -349}\frac{x^2-(-349)^2}{x-(-349)}.$$
(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: $\\boxed{-698}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-698$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Remember: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -698$. |
math-000774 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Work this out carefully: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -397}\frac{x^2-(-397)^2}{x-(-397)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-794}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-794$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -794$. (Here the result is $\boxed{-794}$.) |
math-000775 | Calculus: Limits — Algebraic Simplification | 1 | Work carefully and justify each inference: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -299}\frac{x^2-(-299)^2}{x-(-299)}.$$
(a) Evaluate the limit by algebraic sim... | [
{
"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: $\\boxed{-598}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-598$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robust... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -598$. (Here the result is $\boxed{-598}$.) |
math-000776 | Calculus: Limits — Difference Quotients | 1 | Solve and justify each step: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -340}\frac{x^2-(-340)^2}{x-(-340)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-680}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-680$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -680$. |
math-000777 | Calculus: Limits — Removable Discontinuities | 1 | Exercise: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -235}\frac{x^2-(-235)^2}{x-(-235)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-470}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-470$. 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 $2a= -470$. (Here the result is $\boxed{-470}$.) |
math-000778 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Determine the requested value: Evaluate the limit and explain why direct substitution gives an indeterminate form; then resolve it:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 137}\frac{x^2-(137)^2}{x-(137)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Rei... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{274}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=274$. 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 $2a= 274$. |
math-000779 | Calculus: Limits — Algebraic Simplification | 1 | Checkpoint: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 256}\frac{x^2-(256)^2}{x-(256)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit ... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(256)^2=(x-(256))(x+(256))$.",
"Step 2: For $x\\neq 256$, cancel to get... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{512}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=512$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generali... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 512$. (Here the result is $\boxed{512}$.) |
math-000780 | Calculus: Limits — Removable Discontinuities | 1 | Provide a rigorous solution: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -274}\frac{x^2-(-274)^2}{x-(-274)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-548}$.\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 $a$.",
"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 $2a= -548$. (Here the result is $\boxed{-548}$.) |
math-000781 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | 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:
$$\lim_{x\to -31}\frac{x^2-(-31)^2}{x-(-31)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as ... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-62}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-62$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "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 $2a= -62$. (Here the result is $\boxed{-62}$.) |
math-000782 | Calculus: Limits — Difference Quotients | 1 | Track units/moduli carefully: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -353}\frac{x^2-(-353)^2}{x-(-353)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-706}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-706$. 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 $2a= -706$. (Here the result is $\boxed{-706}$.) |
math-000783 | Calculus: Limits — Difference Quotients | 1 | Proceed methodically: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -152}\frac{x^2-(-152)^2}{x-(-152)}.$$
(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: $\\boxed{-304}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-304$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -304$. |
math-000784 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Work this out carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -350}\frac{x^2-(-350)^2}{x-(-350)}.$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-700}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-700$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -700$. (Here the result is $\boxed{-700}$.) |
math-000785 | Calculus: Limits — Removable Discontinuities | 1 | Problem: 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:
$$\lim_{x\to -288}\frac{x^2-(-288)^2}{x-(-288)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivati... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-576}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-576$. 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 $2a= -576$. (Here the result is $\boxed{-576}$.) |
math-000786 | Calculus: Limits — Removable Discontinuities | 1 | Solve with verification: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -164}\frac{x^2-(-164)^2}{x-(-164)}.$$
(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: $x^2-(-164)^2=(x-(-164))(x+(-164))$.",
"Step 2: For $x\\neq -164$, cancel to... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{-328}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-328$. 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 $2a= -328$. (Here the result is $\boxed{-328}$.) |
math-000787 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Track units/moduli carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 28}\frac{x^2-(28)^2}{x-(28)}.$$
(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: $x^2-(28)^2=(x-(28))(x+(28))$.",
"Step 2: For $x\\neq 28$, cancel to get $\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{56}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=56$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustn... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 56$. |
math-000788 | Calculus: Limits — Difference Quotients | 1 | 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:
$$\lim_{x\to 26}\frac{x^2-(26)^2}{x-(26)}.$$
(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: $x^2-(26)^2=(x-(26))(x+(26))$.",
"Step 2: For $x\\neq 26$, cancel to get $\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. 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 $a$.",
"robustn... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Core principle: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= 52$. (Here the result is $\boxed{52}$.) |
math-000789 | Calculus: Limits — Algebraic Simplification | 1 | Explain each transformation: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 303}\frac{x^2-(303)^2}{x-(303)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) ... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{606}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=606$. 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 $2a= 606$. (Here the result is $\boxed{606}$.) |
math-000790 | Calculus: Limits — Removable Discontinuities | 1 | Challenge: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -382}\frac{x^2-(-382)^2}{x-(-382)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) 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: $\\boxed{-764}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-764$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -764$. |
math-000791 | Calculus: Limits — Removable Discontinuities | 1 | Give an answer and a quick verification: Determine the limit by removing the removable singularity, then confirm via the derivative of $x^2$:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to 216}\frac{x^2-(216)^2}{x-(216)}.$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{432}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=432$. 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 $2a= 432$. |
math-000792 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | 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:
$$\lim_{x\to -45}\frac{x^2-(-45)^2}{x-(-45)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpr... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-45)^2=(x-(-45))(x+(-45))$.",
"Step 2: For $x\\neq -45$, cancel to get... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-90}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-90$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_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 $2a= -90$. |
math-000793 | Calculus: Limits — Difference Quotients | 1 | Work carefully and justify each inference: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -219}\frac{x^2-(-219)^2}{x-(-219)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) R... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-438}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-438$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -438$. (Here the result is $\boxed{-438}$.) |
math-000794 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | 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:
$$\lim_{x\to -291}\frac{x^2-(-291)^2}{x-(-291)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the l... | [
{
"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: $x^2-(-291)^2=(x-(-291))(x+(-291))$.",
"Step 2: For $x\\neq -291$, cancel to... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{-582}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-582$. 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 $2a= -582$. (Here the result is $\boxed{-582}$.) |
math-000795 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Challenge: 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:
$$\lim_{x\to 88}\frac{x^2-(88)^2}{x-(88)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative a... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{176}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=176$. 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 $2a= 176$. |
math-000796 | Calculus: Limits — Algebraic Simplification | 1 | Task: Compute the limit (show two methods) and briefly reconcile them:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -108}\frac{x^2-(-108)^2}{x-(-108)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivative and compute it that way... | [
{
"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: $\\boxed{-216}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-216$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -216$. |
math-000797 | Calculus: Limits — Algebraic Simplification | 1 | Use two approaches if possible: Compute the limit. First simplify algebraically; then interpret it as a derivative (difference quotient):
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -33}\frac{x^2-(-33)^2}{x-(-33)}.$$
(a) Evaluate the limit by algebraic simplification.
(... | [
{
"method_name": "Graph/Linearization Cross-Check",
"approach": "Use that near $x=a$, $x^2$ is approximated by its tangent line with slope $2a$, so the secant slopes approach $2a$.",
"steps": [
"Step 1: The expression is the slope of the secant line through $(x,x^2)$ and $(a,a^2)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{-66}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-66$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"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 $2a= -66$. |
math-000798 | Calculus: Limits — Removable Discontinuities | 1 | Track units/moduli carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -213}\frac{x^2-(-213)^2}{x-(-213)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret th... | [
{
"method_name": "Factor + Cancel (Removable Singularity)",
"approach": "Factor the numerator, cancel the common factor for $x\\neq a$, then take the limit of the simplified expression.",
"steps": [
"Step 1: Factor: $x^2-(-213)^2=(x-(-213))(x+(-213))$.",
"Step 2: For $x\\neq -213$, cancel to... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-426}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-426$. The derivative definition gives the same value because this limit is exactly the derivative of $x^2$ at $a$.",
"robustness_analysis": "Robust... | [
{
"error_description": "Declared the limit 'does not exist' because substituting gives $0/0$.",
"why_plausible": "Students often think 'undefined' implies 'no limit'.",
"why_wrong": "$0/0$ is an indeterminate form that signals simplification or a theorem is needed; many such limits exist.",
"which_m... | Takeaway: A $0/0$ form often indicates a removable discontinuity: factor/cancel or recognize a derivative difference quotient. Here the limit equals $2a= -426$. (Here the result is $\boxed{-426}$.) |
math-000799 | Calculus: Limits — Indeterminate Forms (0/0) | 1 | Problem: 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:
$$\lim_{x\to -124}\frac{x^2-(-124)^2}{x-(-124)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret the limit as a derivati... | [
{
"method_name": "Derivative Definition",
"approach": "Recognize the expression as the difference quotient for $f(x)=x^2$ at $x=a$; by definition it equals $f'(a)$.",
"steps": [
"Step 1: Let $f(x)=x^2$. Then $\\frac{f(x)-f(a)}{x-a}$ is the difference quotient.",
"Step 2: By definition of der... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{-248}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-248$. 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 $2a= -248$. |
math-000800 | Calculus: Limits — Secant-to-Tangent Interpretation | 1 | Track units/moduli carefully: Find the value of the limit using (i) factor/cancel and (ii) derivative interpretation:
Compute the limit and explain why the expression is indeterminate at the point:
$$\lim_{x\to -212}\frac{x^2-(-212)^2}{x-(-212)}.$$
(a) Evaluate the limit by algebraic simplification.
(b) Reinterpret th... | [
{
"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: $\\boxed{-424}$.\nAfter cancellation the limit becomes $\\lim_{x\\to a}(x+a)=2a=-424$. 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 $2a= -424$. (Here the result is $\boxed{-424}$.) |
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