AP Calculus AB Evaluating Limits Algebraically Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Evaluating Limits Algebraically – Maths PDF

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This limit algebra test answers for AP Calculus AB covers evaluating limits algebraically (Common Core AP CALC AB 1.5, AP CALC AB 1.6, AP CALC AB 1.7, AP CALC AB 1.8), with a free PDF to print.

Use this answer key to score the limit algebra test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State Standards: AP CALC AB 1.5, AP CALC AB 1.6, AP CALC AB 1.7, AP CALC AB 1.8

Limit algebra test answers: worked solutions

Question 1 – Substitution and limit laws [2 pts]

$(-1)^{3}-4(-1)+2=-1+4+2=5$. $5$
$\frac{6}{3}+2(3)=2+6=8$. $8$

Scoring: 1 point per part.

Question 2 – Factoring [2 pts]

Substitution gives $\frac{0}{0}$. Factor: $\frac{(x-5)(x+2)}{x-5}=x+2$. The limit is $7$.

Final answer: $7$.

Scoring: 1 point for factoring and canceling; 1 point for the correct limit.

Question 3 – Rationalizing [3 pts]

Multiply by the conjugate: $\frac{(x+25)-25}{x(\sqrt{x+25}+5)}=\frac{1}{\sqrt{x+25}+5}$. Substitute $x=0$: $\frac{1}{10}$.

Final answer: $\frac{1}{10}$.

Scoring: 1 point for identifying the conjugate; 1 point for simplifying; 1 point for the limit.

Question 4 – Trigonometric limit [2 pts]

$\frac{\sin 6x}{2x}=3\cdot\frac{\sin 6x}{6x}\to 3\cdot 1=3$.

Final answer: $3$.

Scoring: 1 point for rewriting with 6x in the denominator; 1 point for the limit.

Question 5 – Squeeze Theorem [3 pts]

$\lim_{x\to 0}(2-x^{2})=2$ and $\lim_{x\to 0}(2+x^{2})=2$. Both bounds have the same limit.

By the Squeeze Theorem, $\lim_{x\to 0} h(x)=2$.

A function between 2 − x² and 2 + x².

Scoring: 1 point for each bound limit; 1 point for the conclusion naming the Squeeze Theorem.

Question 6 – Difference quotient limit [2 pts]

$f(1)=3$ and $f(1+h)=2(1+2h+h^{2})+1+h=3+5h+2h^{2}$.

$\frac{5h+2h^{2}}{h}=5+2h\to 5$. Final answer: $5$.

Scoring: 1 point for expanding and simplifying the quotient; 1 point for the limit.

Question 7 – Hole in a graph [3 pts]

No. The denominator is $0$ at $x=2$.
$\frac{(x+3)(x-2)}{x-2}=x+3$, so the limit is $5$.
The graph is the line $y=x+3$ with a hole at $(2,5)$; the outputs approach height $5$ from both sides.

The graph of g(x) = (x² + x − 6)/(x − 2).

Scoring: 1 point per part.

Question 8 – Cost per item [3 pts]

$\frac{144-100}{2}=22$, so \$22.
$\frac{(n-10)(n+10)}{n-10}=n+10$, so the limit is \$20.

Scoring: 1 point for (a); 1 point for factoring in (b); 1 point for the limit in (b).

Total score

Total

20 points

Key facts behind this limit algebra test answers

Key facts

Direct substitution works when the function is defined and continuous at the target.
The form 0/0 means “simplify more”, not “the limit is 0” and not “does not exist”.
Multiply by the conjugate to remove a square root from a limit.
lim sin(x)/x = 1 and lim (1 – cos x)/x = 0 as x approaches 0 (radians).

How to use this limit algebra test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All AP Calculus AB chapters: AP Calculus AB math.

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