AP Calculus AB — Chapter 1: Limits & Continuity

Show all supporting work on scratch paper. On the AP Exam, Section I is split into a no-calculator and a calculator-allowed part — each question below is labeled accordingly.

Q1EASY 1.3 Estimating from TablesNo Calculator

The table gives values of $f(x)$ near $x=2$.

$x$	1.9	1.99	1.999	2.001	2.01	2.1
$f(x)$	4.61	4.9601	4.996	5.004	5.0401	5.41

Based on the table, $\displaystyle\lim_{x\to 2}f(x)$ is best estimated by

(A) $4$
(B) $4.5$
(C) $5$
(D) does not exist

Q2EASY 1.5 Algebraic ManipulationNo Calculator

$\displaystyle\lim_{x\to 3}\dfrac{x^2-9}{x-3}=$

(A) $0$
(B) $3$
(C) $6$
(D) does not exist

Q3EASY 1.6 RationalizingNo Calculator

$\displaystyle\lim_{x\to 4}\dfrac{\sqrt{x}-2}{x-4}=$

(A) $0$
(B) $\dfrac{1}{4}$
(C) $\dfrac{1}{2}$
(D) $1$

Q4MEDIUM 1.8 Special Trig LimitsNo Calculator

$\displaystyle\lim_{x\to 0}\dfrac{\sin(5x)}{3x}=$

(A) $0$
(B) $\dfrac{3}{5}$
(C) $\dfrac{5}{3}$
(D) does not exist

Q5MEDIUM 1.8 Trig LimitsNo Calculator

$\displaystyle\lim_{x\to 0}\dfrac{1-\cos x}{x^{2}}=$

(A) $0$
(B) $\dfrac{1}{2}$
(C) $1$
(D) does not exist

Q6MEDIUM 1.7 Squeeze TheoremNo Calculator

If $4-x^{2}\le g(x)\le 4+x^{2}$ for all $x$, then $\displaystyle\lim_{x\to 0}g(x)=$

(A) $0$
(B) $2$
(C) $4$
(D) cannot be determined

Q7EASY 1.2 One-Sided from GraphNo Calculator

The graph of $f$ is shown. Which statement is true?

(A) $\displaystyle\lim_{x\to 2^-}f(x)=\lim_{x\to 2^+}f(x)$ and $f(2)$ equals that value.
(B) $\displaystyle\lim_{x\to 2^-}f(x)\ne\lim_{x\to 2^+}f(x)$, so $\displaystyle\lim_{x\to 2}f(x)$ does not exist.
(C) $\displaystyle\lim_{x\to 2}f(x)$ exists and equals $f(2)$.
(D) $\displaystyle\lim_{x\to 2}f(x)$ exists but is not equal to $f(2)$.

Q8MEDIUM 1.9 ContinuityNo Calculator

Let $f(x)=\begin{cases}\dfrac{x^{2}-4}{x-2}, & x\ne 2\\[4pt]k, & x=2\end{cases}$. For what value of $k$ is $f$ continuous at $x=2$?

(A) $0$
(B) $2$
(C) $4$
(D) no such value exists

Q9MEDIUM 1.10 Types of DiscontinuityNo Calculator

$g(x)=\dfrac{x+1}{x^{2}+x}$ has which type of discontinuity at $x=0$?

(A) Removable
(B) Jump
(C) Infinite
(D) $g$ is continuous at $x=0$

Q10HARD 1.11 Piecewise ContinuityNo Calculator

Find $a$ and $b$ so that $f(x)=\begin{cases}2x+a, & x\le 1\\ bx^{2}+3, & 1\lt x\lt 2\\ 4x-b, & x\ge 2\end{cases}$ is continuous everywhere.

(A) $a=2,\ b=1$
(B) $a=3,\ b=2$
(C) $a=1,\ b=0$
(D) No such $(a,b)$ makes $f$ continuous everywhere.

Q11MEDIUM 1.14 Infinite LimitsNo Calculator

$\displaystyle\lim_{x\to 2^-}\dfrac{x+3}{x-2}=$

(A) $+\infty$
(B) $-\infty$
(C) $0$
(D) $1$

Q12MEDIUM 1.15 Limits at InfinityNo Calculator

$\displaystyle\lim_{x\to\infty}\dfrac{6x^{2}-x}{3x^{2}+4}=$

(A) $0$
(B) $1$
(C) $2$
(D) $\infty$

Q13HARD 1.15 Limits at Infinity (Radical)No Calculator

$\displaystyle\lim_{x\to\infty}\dfrac{\sqrt{9x^{4}+1}}{x^{2}-3x}=$

(A) $0$
(B) $1$
(C) $3$
(D) nonexistent

Q14HARD 1.15 End BehaviorNo Calculator

$\displaystyle\lim_{x\to -\infty}\bigl(\sqrt{x^{2}+4x}+x\bigr)=$

(A) $-2$
(B) $0$
(C) $2$
(D) $\infty$

Q15MEDIUM 1.16 IVTNo Calculator

Let $f$ be continuous on $[0,3]$ with $f(0)=-2$ and $f(3)=5$. Which conclusion does the IVT guarantee?

(A) $f(c)=0$ for some $c\in(0,3)$.
(B) $f(c)=6$ for some $c\in(0,3)$.
(C) $f(c)=-3$ for some $c\in(0,3)$.
(D) $f$ is differentiable on $(0,3)$.

Q16HARD 1.16 IVT (Table)No Calculator

The continuous function $h$ has the selected values below. What is the minimum number of real zeros of $h$ on $[1,9]$ guaranteed by the IVT?

$x$	1	3	5	7	9
$h(x)$	$-4$	$2$	$-1$	$3$	$-5$

(A) $1$
(B) $2$
(C) $3$
(D) $4$

Q17MEDIUM 1.12 Intermediate FormsNo Calculator

$\displaystyle\lim_{h\to 0}\dfrac{(2+h)^{3}-8}{h}=$

(A) $0$
(B) $6$
(C) $8$
(D) $12$

Q18HARD 1.13 Complex FractionsNo Calculator

$\displaystyle\lim_{x\to 0}\dfrac{\frac{1}{x+3}-\frac{1}{3}}{x}=$

(A) $-\dfrac{1}{9}$
(B) $0$
(C) $\dfrac{1}{9}$
(D) does not exist

Free-Response Questions

Free-response answers must include complete setup: algebraic manipulation, stated theorem conditions for IVT/Squeeze, and interval justification for continuity. Units and contextual explanations are required where indicated.

FRQ 1EASY 1.5 / 1.6 Evaluating LimitsNo Calculator

Evaluate each limit. Show all algebraic steps.

(a) $\displaystyle\lim_{x\to 5}\dfrac{x^{2}-25}{x^{2}-4x-5}$

(b) $\displaystyle\lim_{x\to 9}\dfrac{x-9}{\sqrt{x}-3}$

FRQ 2MEDIUM 1.9 – 1.11 Continuity & ParametersNo Calculator

Let $f(x)=\begin{cases} \dfrac{x^{2}-x-6}{x-3}, & x<3\\[4pt] ax+b, & 3\le x\le 5\\[4pt] x^{2}-9, & x>5 \end{cases}$.

(a) Find $\displaystyle\lim_{x\to 3^-}f(x)$ and explain how this determines a restriction on $a$ and $b$.

(b) Determine values of $a$ and $b$ that make $f$ continuous on $\mathbb{R}$. Show the system you solve.

(c) With the values from (b), classify the discontinuity of $f'$ at $x=3$ and at $x=5$ (if any). (Preview of Unit 2 — derivative discontinuity. Cram-track students may skip this part.)

FRQ 3MEDIUM 1.14 – 1.15 AsymptotesNo Calculator

Let $f(x)=\dfrac{2x^{2}-x-6}{x^{2}-4}$.

(a) Find all vertical asymptotes of $f$. Justify with one-sided limits.

(b) Find any removable discontinuities and state the value needed to remove each.

(c) Find $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to -\infty}f(x)$, and state the horizontal asymptote.

FRQ 4HARD 1.16 IVT Application (Table)Calculator

A diver's depth $d(t)$ in meters at time $t$ seconds is continuous on $[0,20]$.

$t$ (s)	0	4	10	15	20
$d(t)$ (m)	0	8	22	18	5

(a) Use the IVT to justify that there is a time in $(0,10)$ when the diver is exactly $15$ meters deep.

(b) Is the IVT enough to conclude that the diver is $15$ meters deep at some time in $(10,20)$? Justify.

(c) A student claims the IVT guarantees that the diver's depth equals $25$ meters for some $t\in(0,20)$. Is the claim correct? Explain.

(d) What is the minimum number of times the diver can be at depth $15$ meters on $[0,20]$? Justify.

FRQ 5HARD 1.7 / 1.12 Squeeze & DefinitionNo Calculator

Let $f(x)=x^{2}\cos\!\bigl(\tfrac{1}{x}\bigr)$ for $x\ne 0$, and define $f(0)=0$.

(a) Show, using the Squeeze Theorem, that $\displaystyle\lim_{x\to 0}f(x)=0$. State the bounding inequalities and verify all conditions.

(b) Use part (a) to explain why $f$ is continuous at $x=0$.

(c) Compute $\displaystyle\lim_{x\to 0}\dfrac{f(x)-f(0)}{x-0}$ using the Squeeze Theorem. Interpret the meaning of the limit.

Limits & Continuity

Multiple Choice Questions

Free-Response Questions