AP Calculus AB Infinite Limits and Asymptotes Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Infinite Limits and Asymptotes – Maths PDF

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This asymptotes test answers for AP Calculus AB covers infinite limits and asymptotes (Common Core AP CALC AB 1.13, AP CALC AB 1.14, AP CALC AB 1.15), with a free PDF to print.

Use this answer key to score the asymptotes 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 1.13, AP CALC AB 1.14, AP CALC AB 1.15

Asymptotes test answers: worked solutions

Question 1 – One-sided infinite limits [2 pts]

From the right, $x-5 > 0$ is small, so $\frac{-2}{x-5}\to-\infty$. From the left, $x-5 < 0$, so the quotient goes to $\infty$.

Final answer: $-\infty$ from the right; $\infty$ from the left.

Scoring: 1 point for each one-sided limit.

Question 2 – Vertical asymptotes [2 pts]

$\frac{x-1}{(x-1)(x+1)}=\frac{1}{x+1}$. The vertical asymptote is $x=-1$, and there is a hole at $x=1$ (at $\left(1,\frac{1}{2}\right)$).

Scoring: 1 point for the asymptote; 1 point for identifying the hole.

Question 3 – Limits at infinity [3 pts]

$\frac{7}{2}$
$0$
$\infty$

Reasons: (a) equal degrees give the ratio of leading coefficients; (b) denominator degree is larger; (c) numerator degree is larger.

Scoring: 1 point per limit.

Question 4 – Both directions [2 pts]

For large $|x|$, $\sqrt{x^{2}+1}\approx|x|$. As $x\to\infty$ the quotient approaches $2$; as $x\to-\infty$ it approaches $-2$.

Final answer: $2$ and $-2$.

Scoring: 1 point for each limit.

Question 5 – Reading asymptotes [3 pts]

$\infty$: the outputs grow without bound on both sides.
$1$ and $1$.
$x=-2$ (vertical) and $y=1$ (horizontal).

Scoring: 1 point per part.

Question 6 – All asymptotes [3 pts]

Vertical asymptote: $x=2$ (numerator $9\neq 0$). Horizontal asymptote: $\frac{3}{1}=3$, so $y=3$.

x-intercept: $3x+3=0$ gives $(-1,0)$. y-intercept: $r(0)=\frac{3}{-2}=-1.5$, so $(0,-1.5)$.

The graph of r(x) = 3(x + 1)/(x − 2).

Scoring: 1 point for each asymptote; 1 point for both intercepts.

Question 7 – Average cost [2 pts]

$\frac{90+90}{30}=6$, so \$6 per loaf.
$A(n)=3+\frac{90}{n}\to 3$: the average cost approaches \$3 per loaf, the cost of making one more loaf.

Scoring: 1 point per part.

Question 8 – Leveling off [3 pts]

$\frac{320}{8}=40$ rpm.
$80$: equal degrees give $\frac{80}{1}$.
The fan speed approaches 80 rpm but the model never exceeds it; the line $D=80$ is a horizontal asymptote.

Distance a drone travels, approaching a limit.

Scoring: 1 point per part.

Total score

Total

20 points

Key facts behind this asymptotes test answers

Key facts

A vertical asymptote at x = a occurs when a one-sided limit there is infinite.
A horizontal asymptote y = L comes from a limit at infinity equal to L.
Rational functions: smaller top degree gives 0, equal degrees give the ratio of leading coefficients.
Check the sign on each side of a vertical asymptote to decide between +infinity and -infinity.

How to use this asymptotes 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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