AP Calculus AB Unit 1: Limits and Continuity

What a limit means and how to find one from a table, a formula or algebra; one-sided limits; the squeeze theorem; the three-part test for continuity; removable, jump and infinite discontinuities; limits involving infinity and asymptotes; and the Intermediate Value Theorem.

CED topics in this unit: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 1.10, 1.11, 1.12, 1.13, 1.14, 1.15, 1.16.

Key formulas and facts

• $\lim_{x\to c}f(x)=L$ exists only if $\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=L$
• $\lim_{x\to0}\frac{\sin x}{x}=1$ and $\lim_{x\to0}\frac{1-\cos x}{x}=0$
• Continuity at $x=c$: $f(c)$ is defined, $\lim_{x\to c}f(x)$ exists, and $\lim_{x\to c}f(x)=f(c)$
• Squeeze theorem: if $g(x)\le f(x)\le h(x)$ near $c$ and $\lim_{x\to c}g(x)=\lim_{x\to c}h(x)=L$, then $\lim_{x\to c}f(x)=L$
• Limits at infinity of a rational function: compare the degrees of the numerator and the denominator
• IVT: if $f$ is continuous on $[a,b]$ and $k$ is between $f(a)$ and $f(b)$, then $f(c)=k$ for some $c$ in $(a,b)$

Worked examples

• Factor and cancel: $\lim_{x\to4}\frac{x^2-x-12}{x-4}=\lim_{x\to4}(x+3)=7$
• Conjugate: $\lim_{x\to9}\frac{\sqrt x-3}{x-9}=\lim_{x\to9}\frac{1}{\sqrt x+3}=\frac16$
• Make $f(x)=\begin{cases}kx+1, & x<2\\ x^2-k, & x\ge2\end{cases}$ continuous: $2k+1=4-k$, so $k=1$
• Horizontal asymptote: $\lim_{x\to\infty}\frac{5x^2-3}{2x^2+x}=\frac52$, so $y=\frac52$
• IVT: $g$ is continuous on $[1,3]$ with $g(1)=-2$ and $g(3)=5$. Since $-2<0<5$, $g(c)=0$ for some $c$ in $(1,3)$

Common traps

• Saying a limit exists because $f(c)$ is defined: the limit does not depend on the value at $c$.
• Giving $\frac00$ as the answer: it means more work is needed (factor, use a conjugate, or a known limit).
• Using the IVT without saying that the function is continuous on the closed interval.
• Forgetting that $\sqrt{x^2}=|x|$: $\lim_{x\to-\infty}\frac{\sqrt{x^2+1}}{x}=-1$, not $1$.
• Calling every zero of a denominator a vertical asymptote, even when the factor cancels (that is a removable discontinuity).

Practice questions

113 multiple-choice and 4 free-response questions. Try each one before opening the answer. Each line of a free-response scoring guide is one point. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

What is $\displaystyle\lim_{x\to -2}\frac{x^2+7x+10}{x^2-4}$?

1. $- \frac{3}{4}$
2. $\frac{3}{4}$
3. $0$
4. The limit does not exist.

Question 2 · Multiple choice

What is $\displaystyle\lim_{x\to 0}\frac{\sqrt{x+25}-5}{x}$?

1. $\frac{1}{5}$
2. $5$
3. $0$
4. $\frac{1}{10}$

Question 3 · Multiple choice

For all $x$, the function $f$ satisfies $3x-2\le f(x)\le x^2-x+2$. What is $\displaystyle\lim_{x\to 2}f(x)$?

1. $2$
2. $4$
3. $0$
4. It cannot be determined from the information given.

Question 4 · Multiple choice

The table gives values of a function $h$ for $x$ near $2$. The function is not defined at $x=2$. Based on the table, what is the best estimate of $\displaystyle\lim_{x\to 2}h(x)$?

1. $12$
2. $11.994$
3. $12.006$
4. The limit does not exist because $h(2)$ is not defined.

Question 5 · Multiple choice

Let $f(x)=\begin{cases}ax+4, & x\le 2\\ x^3-2a, & x>2.\end{cases}$ For what value of $a$ is $f$ continuous at $x=2$?

1. $2$
2. $-1$
3. $1$
4. $3$

Question 6 · Multiple choice

Let $g(x)=\dfrac{x^2-9}{x^2-2x-3}$. Which statement describes $g$ at $x=3$ and $x=-1$?

1. Removable discontinuity at $x=3$; vertical asymptote at $x=-1$
2. Vertical asymptotes at both $x=3$ and $x=-1$
3. Removable discontinuities at both $x=3$ and $x=-1$
4. Vertical asymptote at $x=3$; removable discontinuity at $x=-1$

Question 7 · Multiple choice

The function $f$ is continuous on $[0,4]$, with $f(0)=3$, $f(2)=-1$ and $f(4)=2$. Which statement must be true?

1. $f(c)=1$ for exactly two values of $c$ in $(0,4)$
2. $f(x)=0$ has at least two solutions in $(0,4)$
3. The maximum value of $f$ on $[0,4]$ is $3$
4. $f(1)=1$

Question 8 · Multiple choice

What is $\displaystyle\lim_{x\to -\infty}\frac{\sqrt{9x^2+4}}{2x-1}$?

1. $\frac{3}{2}$
2. $- \frac{3}{2}$
3. $\frac{9}{2}$
4. $0$

Question 9 · Free response (9 points)

Let $f(x)=\dfrac{x^2+x-12}{x^2-9}$.

1. (a) Find $\displaystyle\lim_{x\to3}f(x)$. Show the work that leads to your answer. [2 points]
2. (b) For each of $x=3$ and $x=-3$, say what kind of discontinuity $f$ has there. Justify your answers using limits. [2 points]
3. (c) Find $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to-\infty}f(x)$. What do these limits tell you about the graph of $f$? [2 points]
4. (d) Let $g(x)=f(x)$ for $x\ne3$ and $g(3)=c$. Find the value of $c$ for which $g$ is continuous at $x=3$. [1 point]
5. (e) Explain why there is a value $x=k$ with $0<k<2$ for which $g(k)=1.25$. [2 points]

Question 10 · Free response (9 points)

Let $h(x)=\begin{cases}2x+a, & x\le -1\\ x^2+bx+1, & -1<x<2\\ 4x-1, & x\ge 2\end{cases}$ where $a$ and $b$ are constants.

1. (a) Find $\displaystyle\lim_{x\to2^+}h(x)$, and write $\displaystyle\lim_{x\to2^-}h(x)$ in terms of $b$. [2 points]
2. (b) Find the values of $a$ and $b$ for which $h$ is continuous for all real numbers $x$. [3 points]
3. (c) A function $p$ satisfies $1-x^2\le p(x)\le\cos x$ for all $x$. Find $\displaystyle\lim_{x\to0}p(x)$ and justify your answer. [2 points]
4. (d) Without using L'Hospital's rule, find $\displaystyle\lim_{x\to0}\frac{1-\cos(4x)}{x\sin(2x)}$. [2 points]

Question 11 · Multiple choice

The position of a cart on a straight track is $s(t)=t^2+3t$ feet at time $t$ seconds. What is the average velocity of the cart over the time interval $2\le t\le 2.1$?

1. $\frac{71}{10}$
2. $7$
3. $\frac{71}{100}$
4. $\frac{107}{10}$

Question 12 · Multiple choice

For a function $f$, the table gives the average rate of change of $f$ over the interval $[1,\,1+h]$ for three values of $h$. Based on the table, what is the best estimate of the instantaneous rate of change of $f$ at $x=1$?

1. $3.003$
2. $1$
3. $3.310$
4. $3$

Question 13 · Multiple choice

Let $g(x)=\sqrt{x}$. Which of the following gives the instantaneous rate of change of $g$ at $x=9$?

1. $\dfrac{\sqrt{10}-3}{1}$
2. $\displaystyle\lim_{h\to0}\frac{\sqrt{9+h}-3}{h}$
3. $\displaystyle\lim_{h\to0}\left(\sqrt{9+h}-3\right)$
4. $\displaystyle\lim_{h\to0}\frac{\sqrt{9+h}-3}{9+h}$

Question 14 · Multiple choice

The temperature of a cup of coffee is recorded at selected times. Using the average rate of change of $T$ over $2\le t\le 2.5$, what is the estimate of the rate of change of the temperature at $t=2$ minutes, in degrees Fahrenheit per minute?

1. $- \frac{15}{2}$
2. $-6$
3. $-3$
4. $- \frac{12}{5}$

Question 15 · Multiple choice

For $f(x)=\dfrac1x$ and $h\neq0$, which expression is equal to the average rate of change of $f$ over the interval $[2,\,2+h]$?

1. $- \frac{1}{4}$
2. $- \frac{1}{2 h + 4}$
3. $\frac{1}{2 h + 4}$
4. $- \frac{1}{\left(h + 2\right)^{2}}$

Question 16 · Multiple choice

A ball's height is modeled by $y(t)=4t^2$ feet for $0\le t\le 5$ seconds. For $h>0$, the average velocity of the ball over $[3,\,3+h]$ is $24+4h$ feet per second. For what value of $h$ is this average velocity $24.4$ feet per second?

1. $\frac{1}{100}$
2. $\frac{2}{5}$
3. $\frac{1}{10}$
4. $1$

Question 17 · Multiple choice

For $f(x)=x^2-4x$ and $h\neq0$, the average rate of change of $f$ over $[3,\,3+h]$ simplifies to $h+2$. What is the instantaneous rate of change of $f$ at $x=3$?

1. $2$
2. $3$
3. $-3$
4. $0$

Question 18 · Multiple choice

Let $f(x)=e^{0.5x}$. What is the average rate of change of $f$ over the interval $[2,\,2.001]$?

1. $2.718$
2. $0.680$
3. $2.719$
4. $1.359$

Question 19 · Multiple choice

Let $f(x)=\begin{cases}x^2+1, & x<1\\ 5-2x, & x\ge 1.\end{cases}$ What is $\displaystyle\lim_{x\to1^{+}}f(x)$?

1. $2$
2. $3$
3. $\frac{5}{2}$
4. The limit does not exist.

Question 20 · Multiple choice

What is $\displaystyle\lim_{x\to2^{-}}\frac{x^2-4}{|x-2|}$?

1. $4$
2. $0$
3. $-4$
4. The limit does not exist.

Question 21 · Multiple choice

For which of the following functions does $\displaystyle\lim_{x\to3}f(x)$ NOT exist?

1. $f(x)=\dfrac{x-3}{|x-3|}$
2. $f(x)=\dfrac{x^2-9}{x-3}$
3. $f(x)=\dfrac{\sin(x-3)}{x-3}$
4. $f(x)=\sqrt{x+1}$

Question 22 · Multiple choice

The function $f$ is defined by $f(x)=\dfrac{x^2+x-6}{x-2}$ for $x\neq2$ and $f(2)=1$. Which of the following is true?

1. $\displaystyle\lim_{x\to2}f(x)=1$ and $f(2)=1$
2. $\displaystyle\lim_{x\to2}f(x)=5$ and $f(2)=1$
3. $\displaystyle\lim_{x\to2}f(x)=5$ and $f(2)=5$
4. $\displaystyle\lim_{x\to2}f(x)$ does not exist and $f(2)=1$

Question 23 · Multiple choice

What is $\displaystyle\lim_{x\to4^{-}}\left(x+\sqrt{4-x}\right)$?

1. $4$
2. $0$
3. $6$
4. The limit does not exist.

Question 24 · Multiple choice

For a function $f$, $\displaystyle\lim_{x\to2^{-}}f(x)=3$ and $\displaystyle\lim_{x\to2^{+}}f(x)=3$. Which of the following must be true?

1. $f(2)=3$
2. $\displaystyle\lim_{x\to2}f(x)=3$
3. $f$ is continuous at $x=2$
4. $f(2)$ is defined

Question 25 · Multiple choice

Let $f(x)=\begin{cases}\dfrac{x^2-1}{x-1}, & x<1\\ ax+1, & x>1.\end{cases}$ For what value of $a$ does $\displaystyle\lim_{x\to1}f(x)$ exist?

1. $2$
2. $0$
3. $1$
4. $3$

Question 26 · Multiple choice

Which of the following limits does NOT exist?

1. $\displaystyle\lim_{x\to0}x\sin\!\left(\frac1x\right)$
2. $\displaystyle\lim_{x\to0}\cos x$
3. $\displaystyle\lim_{x\to0}\sin\!\left(\frac1x\right)$
4. $\displaystyle\lim_{x\to0}\frac{\sin x}{x}$

Question 27 · Multiple choice

The graph of the function $f$ is shown. What is $\displaystyle\lim_{x\to1}f(x)$?

1. $4$
2. $2$
3. $3$
4. The limit does not exist.

Question 28 · Multiple choice

The graph of the function $f$ is shown. For which value of $c$ does $\displaystyle\lim_{x\to c}f(x)$ exist but is not equal to $f(c)$?

1. $-2$
2. $3$
3. $1$
4. $0$

Question 29 · Multiple choice

The graph of $f$ is shown, with vertical asymptote $x=2$. What is $\displaystyle\lim_{x\to2^{+}}f(x)$?

1. $\infty$
2. $-\infty$
3. $1$
4. $3$

Question 30 · Multiple choice

The graph of $f$ is shown, with vertical asymptote $x=2$. What is $\displaystyle\lim_{x\to0}f(x)+\lim_{x\to4}f(x)$?

1. $2$
2. $3$
3. $1$
4. The limit does not exist.

Question 31 · Multiple choice

The graph of $f$ is shown. What is $\displaystyle\lim_{x\to1^{-}}f(x)$?

1. $- \frac{1}{2}$
2. $2$
3. $\frac{1}{2}$
4. The limit does not exist.

Question 32 · Multiple choice

The graph of $f$ is shown. What is $\displaystyle\lim_{x\to1^{+}}f(x)-\lim_{x\to1^{-}}f(x)$?

1. $\frac{5}{2}$
2. $\frac{3}{2}$
3. $- \frac{5}{2}$
4. $0$

Question 33 · Multiple choice

The graph of $f$ consists of a semicircle and a line segment, as shown. What is $\displaystyle\lim_{x\to2^{+}}f(x)$?

1. $0$
2. $2$
3. $1$
4. The limit does not exist.

Question 34 · Multiple choice

The graph of $f$ consists of a semicircle and a line segment, as shown. What is $\displaystyle\lim_{x\to0}f(x)$?

1. $0$
2. $2$
3. $4$
4. The limit does not exist.

Question 35 · Multiple choice

The table gives values of a function $g$ for $x$ near $0$; $g(0)$ is not defined. Based on the table, what is the best estimate of $\displaystyle\lim_{x\to0}g(x)$?

1. $1$
2. $0$
3. $2$
4. $\frac{11}{5}$

Question 36 · Multiple choice

The table gives values of a function $h$ for $x$ near $1$. Which statement is best supported by the table?

1. $\displaystyle\lim_{x\to1}h(x)$ does not exist.
2. $\displaystyle\lim_{x\to1}h(x)=3$
3. $\displaystyle\lim_{x\to1}h(x)=-1$
4. $\displaystyle\lim_{x\to1}h(x)=1$

Question 37 · Multiple choice

The table gives values of a function $f$ for $x$ near $2$. Based on the table, what is the best estimate of $\displaystyle\lim_{x\to2^{+}}f(x)$?

1. $4$
2. $\frac{41}{10}$
3. The limit does not exist.
4. $5$

Question 38 · Multiple choice

The table gives values of a function $k$ for $x$ near $3$. Which statement is best supported by the table?

1. $\displaystyle\lim_{x\to3}k(x)=0$
2. $\displaystyle\lim_{x\to3}k(x)=1000000$
3. $\displaystyle\lim_{x\to3}k(x)=\infty$
4. $\displaystyle\lim_{x\to3^-}k(x)=-\infty$

Question 39 · Multiple choice

The table gives values of a function $f$ for $x$ near $3$, including $x=3$. Based on the table, what is the best estimate of $\displaystyle\lim_{x\to3}f(x)$?

1. $7$
2. $6$
3. $5$
4. The limit does not exist.

Question 40 · Multiple choice

The table gives values of $r(x)$ for $x$ near $0$ ($r(0)$ is not defined). Based on the table, what is the best estimate of $\displaystyle\lim_{x\to0}r(x)$?

1. $1$
2. $0$
3. $3$
4. $\frac{13}{5}$

Question 41 · Multiple choice

The table gives values of functions $f$ and $g$ for $x$ near $2$. Based on the table, what is the best estimate of $\displaystyle\lim_{x\to2}\big(f(x)\,g(x)\big)$?

1. $4$
2. $-4$
3. $3$
4. $-2$

Question 42 · Multiple choice

If $\displaystyle\lim_{x\to4}f(x)=3$ and $\displaystyle\lim_{x\to4}g(x)=-2$, what is $\displaystyle\lim_{x\to4}\left[2f(x)-\big(g(x)\big)^2\right]$?

1. $10$
2. $-2$
3. $8$
4. $2$

Question 43 · Multiple choice

If $\displaystyle\lim_{x\to c}f(x)=5$ and $\displaystyle\lim_{x\to c}g(x)=0$, what is $\displaystyle\lim_{x\to c}\frac{f(x)+3}{g(x)-2}$?

1. $4$
2. $- \frac{5}{2}$
3. $-4$
4. The limit does not exist.

Question 44 · Multiple choice

If $\displaystyle\lim_{x\to1}f(x)=4$, what is $\displaystyle\lim_{x\to1}\left(\sqrt{f(x)}+x^2\right)$?

1. $3$
2. $5$
3. $17$
4. $2$

Question 45 · Multiple choice

For functions $f$ and $g$, $\displaystyle\lim_{x\to2^-}f(x)=1$, $\displaystyle\lim_{x\to2^+}f(x)=3$ and $\displaystyle\lim_{x\to2}g(x)=2$. What is $\displaystyle\lim_{x\to2^+}\big(f(x)\,g(x)\big)$?

1. $2$
2. $6$
3. $8$
4. The limit does not exist.

Question 46 · Multiple choice

If $\displaystyle\lim_{x\to3}f(x)=2$ and $g(u)=u^2-5$, what is $\displaystyle\lim_{x\to3}g\big(f(x)\big)$?

1. $-1$
2. $4$
3. $2$
4. $-5$

Question 47 · Multiple choice

If $\displaystyle\lim_{x\to0}f(x)=-1$, what is $\displaystyle\lim_{x\to0}\left[\big(f(x)\big)^3-2f(x)\right]$?

1. $1$
2. $-3$
3. $3$
4. $-1$

Question 48 · Multiple choice

For a function $f$, $\displaystyle\lim_{x\to2}\frac{f(x)}{x-2}=5$. What is $\displaystyle\lim_{x\to2}f(x)$?

1. $5$
2. $0$
3. $10$
4. The limit does not exist.

Question 49 · Multiple choice

What is $\displaystyle\lim_{x\to3}\frac{x^3-27}{x-3}$?

1. $9$
2. $27$
3. $0$
4. The limit does not exist.

Question 50 · Multiple choice

What is $\displaystyle\lim_{x\to-1}\frac{x^2-1}{x^2+3x+2}$?

1. $2$
2. $0$
3. The limit does not exist.
4. $-2$

Question 51 · Multiple choice

What is $\displaystyle\lim_{x\to4}\frac{\frac1x-\frac14}{x-4}$?

1. $\frac{1}{16}$
2. $- \frac{1}{16}$
3. $- \frac{1}{4}$
4. $0$

Question 52 · Multiple choice

What is $\displaystyle\lim_{x\to2}\frac{x^4-16}{x-2}$?

1. $32$
2. $16$
3. $8$
4. $0$

Question 53 · Multiple choice

What is $\displaystyle\lim_{x\to2}\frac{x^2-7x+10}{x^2-4}$?

1. $\frac{3}{4}$
2. $- \frac{5}{2}$
3. $- \frac{3}{4}$
4. The limit does not exist.

Question 54 · Multiple choice

What is $\displaystyle\lim_{x\to0}\frac{\sin(5x)}{2x}$?

1. $\frac{2}{5}$
2. $5$
3. $\frac{1}{2}$
4. $\frac{5}{2}$

Question 55 · Multiple choice

What is $\displaystyle\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}$?

1. $\frac{1}{3}$
2. $6$
3. $0$
4. $\frac{1}{6}$

Question 56 · Multiple choice

What is $\displaystyle\lim_{x\to0}\frac{\tan(3x)}{\sin(2x)}$?

1. $\frac{2}{3}$
2. $1$
3. $6$
4. $\frac{3}{2}$

Question 57 · Multiple choice

What is $\displaystyle\lim_{x\to0}\frac{x}{\sqrt{x+4}-2}$?

1. $\frac{1}{4}$
2. $2$
3. $0$
4. $4$

Question 58 · Multiple choice

What is $\displaystyle\lim_{\theta\to0}\frac{1-\cos(2\theta)}{\theta^2}$?

1. $1$
2. $\frac{1}{2}$
3. $2$
4. $0$

Question 59 · Multiple choice

What is $\displaystyle\lim_{x\to2}\frac{x\,|x-2|}{x-2}$?

1. $2$
2. $-2$
3. $0$
4. The limit does not exist.

Question 60 · Multiple choice

What is $\displaystyle\lim_{x\to1}\frac{\sqrt{x+3}-2}{x^2-1}$?

1. $\frac{1}{4}$
2. $\frac{1}{8}$
3. $\frac{1}{2}$
4. $0$

Question 61 · Multiple choice

What is $\displaystyle\lim_{x\to0}x^2\cos\!\left(\frac{3}{x}\right)$?

1. $1$
2. $3$
3. $0$
4. The limit does not exist.

Question 62 · Multiple choice

A function $g$ satisfies $4x-x^2\le g(x)\le x^2-4x+8$ for all $x$. What is $\displaystyle\lim_{x\to2}g(x)$?

1. $8$
2. $0$
3. $4$
4. It cannot be determined.

Question 63 · Multiple choice

What is $\displaystyle\lim_{x\to\infty}\frac{\sin x}{x}$?

1. $1$
2. $0$
3. $-1$
4. The limit does not exist.

Question 64 · Multiple choice

For $0<|x|<1$, $1-\dfrac{x^2}{6}\le\dfrac{\sin x}{x}\le1$. Which value does the squeeze theorem give for $\displaystyle\lim_{x\to0}\frac{\sin x}{x}$?

1. $0$
2. $\frac{5}{6}$
3. $1$
4. $\frac{1}{6}$

Question 65 · Multiple choice

A function $f$ satisfies $|f(x)-3|\le2|x-1|$ for all $x$. What is $\displaystyle\lim_{x\to1}f(x)$?

1. $1$
2. $3$
3. $2$
4. It cannot be determined.

Question 66 · Multiple choice

For $x$ near $0$, a function $g$ satisfies $\cos x\le g(x)\le1+x^2$. What is $\displaystyle\lim_{x\to0}g(x)$?

1. $0$
2. $1$
3. $2$
4. It cannot be determined.

Question 67 · Multiple choice

The graph of $f$ is shown. What is $\displaystyle\lim_{x\to1}f\big((x-1)^2\big)$?

1. $-1$
2. $1$
3. $3$
4. The limit does not exist.

Question 68 · Multiple choice

The graph of $f$ is shown, and $g(x)=x^2+1$. What is $\displaystyle\lim_{x\to0^{-}}\big(g(x)-f(x)\big)$?

1. $-2$
2. $0$
3. $2$
4. $4$

Question 69 · Multiple choice

The graph of $f$ is shown. What is $\displaystyle\lim_{x\to0}x\,f(x)$?

1. $3$
2. $-1$
3. The limit does not exist.
4. $0$

Question 70 · Multiple choice

The table gives values of a function $f$ for $x$ near $3$. Based on the table, what is the best estimate of $\displaystyle\lim_{x\to1.5}f(2x)$?

1. $3$
2. $6$
3. $12$
4. The limit does not exist.

Question 71 · Multiple choice

The function $f$ is continuous at $x=2$ with $f(2)=5$, and $g(x)=\dfrac{x^2-4}{x-2}$ for $x\neq2$. What is $\displaystyle\lim_{x\to2}f(x)\,g(x)$?

1. $20$
2. $5$
3. $4$
4. The limit does not exist.

Question 72 · Multiple choice

The graph of $f$ consists of a semicircle and a line segment, as shown. What is $\displaystyle\lim_{x\to1^{+}}f(x+1)$?

1. $0$
2. $1$
3. $2$
4. The limit does not exist.

Question 73 · Multiple choice

For $x\neq2$, $f(x)=\dfrac{x^2+kx-10}{x-2}$, where $k$ is a constant, and $\displaystyle\lim_{x\to2}f(x)$ exists. What is the value of the limit?

1. $3$
2. $5$
3. $7$
4. $2$

Question 74 · Multiple choice

Let $f(x)=\begin{cases}x+1, & x<2\\ x^2, & x\ge2.\end{cases}$ Which describes $f$ at $x=2$?

1. jump discontinuity
2. removable discontinuity
3. infinite discontinuity
4. continuous

Question 75 · Multiple choice

Let $f(x)=\dfrac{x+2}{x^2-4}$. Which describes $f$ at $x=2$ and at $x=-2$?

1. Removable discontinuity at $x=2$; infinite discontinuity at $x=-2$
2. Infinite discontinuity at $x=2$; removable discontinuity at $x=-2$
3. Infinite discontinuities at both
4. Jump discontinuities at both

Question 76 · Multiple choice

Let $g(x)=\dfrac{x^2-4x}{|x-4|}$. Which describes $g$ at $x=4$?

1. removable discontinuity
2. infinite discontinuity
3. continuous
4. jump discontinuity

Question 77 · Multiple choice

The graph of $f$ is shown. At which value of $x$ does $f$ have a removable discontinuity?

1. $1$
2. $-2$
3. $3$
4. $0$

Question 78 · Multiple choice

Let $h(x)=\dfrac{\sin x}{x}$ for $x\neq0$, with $h(0)$ not defined. Which describes $h$ at $x=0$?

1. jump discontinuity
2. removable discontinuity
3. infinite discontinuity
4. continuous

Question 79 · Multiple choice

Let $p(x)=\dfrac{x-1}{x^2-1}$. Which describes $p$ at $x=-1$?

1. removable discontinuity
2. jump discontinuity
3. infinite discontinuity
4. continuous

Question 80 · Multiple choice

The function $f$ is given by $f(x)=kx^2+1$ for $x<3$ and $f(x)=4x-k$ for $x\ge3$, where $k$ is a constant. For what value of $k$ is $f$ continuous at $x=3$?

1. $\frac{11}{10}$
2. $1$
3. $- \frac{11}{10}$
4. $\frac{11}{8}$

Question 81 · Multiple choice

Let $f(x)=\dfrac{x^2-16}{x-4}$ for $x\neq4$ and $f(4)=c$. For what value of $c$ is $f$ continuous at $x=4$?

1. $4$
2. $8$
3. $0$
4. $16$

Question 82 · Multiple choice

Let $f(x)=x^2$ for $x\neq2$ and $f(2)=3$. Which condition for continuity at $x=2$ fails?

1. $f(2)$ is not defined
2. $\displaystyle\lim_{x\to2}f(x)\neq f(2)$
3. $\displaystyle\lim_{x\to2}f(x)$ does not exist
4. None: $f$ is continuous at $x=2$

Question 83 · Multiple choice

Let $f(x)=\begin{cases}ax+b, & x<1\\ 4, & x=1\\ x^2+a, & x>1.\end{cases}$ If $f$ is continuous at $x=1$, what is the value of $b$?

1. $3$
2. $4$
3. $1$
4. $0$

Question 84 · Multiple choice

The graph of $f$ is shown. At which value of $x$ is $f$ continuous?

1. $0$
2. $-2$
3. $1$
4. $3$

Question 85 · Multiple choice

Let $f(x)=\dfrac{\sqrt{x+5}-3}{x-4}$ for $x\neq4$ and $f(4)=k$. For what value of $k$ is $f$ continuous at $x=4$?

1. $\frac{1}{3}$
2. $3$
3. $\frac{1}{6}$
4. $0$

Question 86 · Multiple choice

On which set is $f(x)=\dfrac{\sqrt{x-2}}{x-5}$ continuous?

1. $(2,\infty)$
2. $[2,\infty)$
3. $(-\infty,5)\cup(5,\infty)$
4. $[2,5)\cup(5,\infty)$

Question 87 · Multiple choice

On which set is $g(x)=\ln(x^2-9)$ continuous?

1. $(3,\infty)$
2. $(-3,3)$
3. $(-\infty,\infty)$
4. $(-\infty,-3)\cup(3,\infty)$

Question 88 · Multiple choice

At how many values of $x$ in the open interval $(0,\,2\pi)$ is $f(x)=\sec x$ NOT continuous?

1. $0$
2. $2$
3. $1$
4. $4$

Question 89 · Multiple choice

At which values of $x$ is $h(x)=\dfrac{x-1}{x^2+x-6}$ NOT continuous?

1. $x=3$ and $x=-2$ only
2. $x=1$ only
3. $x=-3$ and $x=2$ only
4. $x=-3$, $x=1$ and $x=2$

Question 90 · Multiple choice

Let $f(x)=\begin{cases}e^x, & x\le0\\ x+1, & x>0.\end{cases}$ Which statement is true?

1. f has a jump discontinuity at x = 0
2. f is continuous for all real x
3. f has a removable discontinuity at x = 0
4. f has an infinite discontinuity at x = 0

Question 91 · Multiple choice

On which set is $f(x)=\dfrac{x}{e^x-1}$ continuous?

1. $(-\infty,\infty)$
2. $(-\infty,0)\cup(0,\infty)$
3. $(0,\infty)$
4. $(-\infty,1)\cup(1,\infty)$

Question 92 · Multiple choice

On which set is $f(x)=\sqrt{16-x^2}$ continuous?

1. $(-4,4)$
2. $[0,4]$
3. $(-\infty,\infty)$
4. $[-4,4]$

Question 93 · Multiple choice

The function $f(x)=\dfrac{x^2+2x-15}{x-3}$ is not defined at $x=3$. What value should be assigned to $f(3)$ to make $f$ continuous at $x=3$?

1. $8$
2. $5$
3. $3$
4. $0$

Question 94 · Multiple choice

Let $g(x)=\dfrac{\sin(4x)}{x}$ for $x\neq0$. What value of $g(0)$ makes $g$ continuous at $x=0$?

1. $4$
2. $1$
3. $\frac{1}{4}$
4. $0$

Question 95 · Multiple choice

Let $h(x)=\dfrac{x-9}{\sqrt{x}-3}$ for $x\ge0$, $x\neq9$. What value of $h(9)$ makes $h$ continuous at $x=9$?

1. $3$
2. $\frac{1}{6}$
3. $6$
4. $9$

Question 96 · Multiple choice

Let $f(x)=\dfrac{x^3+1}{x+1}$ for $x\neq-1$. What value of $f(-1)$ removes the discontinuity?

1. $3$
2. $1$
3. $0$
4. $-1$

Question 97 · Multiple choice

Let $f(x)=\begin{cases}x^2+k, & x<2\\ 3x-k, & x\ge2.\end{cases}$ For what value of $k$ is $f$ continuous for all $x$?

1. $1$
2. $2$
3. $-1$
4. $5$

Question 98 · Multiple choice

Let $f(x)=\dfrac{e^x-1}{2x}$ for $x\neq0$. What value of $f(0)$ makes $f$ continuous at $x=0$?

1. $1$
2. $2$
3. $0$
4. $\frac{1}{2}$

Question 99 · Multiple choice

What is $\displaystyle\lim_{x\to3^{-}}\frac{x+1}{x-3}$?

1. $\infty$
2. $-\infty$
3. $- \frac{4}{3}$
4. $0$

Question 100 · Multiple choice

What is $\displaystyle\lim_{x\to-2^{+}}\frac{x}{(x+2)^2}$?

1. $\infty$
2. $- \frac{1}{2}$
3. $-\infty$
4. $0$

Question 101 · Multiple choice

Let $r(x)=\dfrac{x^2-9}{x^2+x-6}$. Which lines are vertical asymptotes of the graph of $r$?

1. $x=2$ and $x=-3$
2. $x=-3$ only
3. $x=2$ and $x=3$
4. $x=2$ only

Question 102 · Multiple choice

What is $\displaystyle\lim_{x\to\pi^{-}}\cot x$?

1. $\infty$
2. $-1$
3. $0$
4. $-\infty$

Question 103 · Multiple choice

What is $\displaystyle\lim_{x\to2}\frac{1}{x^2-4x+4}$?

1. $-\infty$
2. $\frac{1}{4}$
3. $0$
4. $\infty$

Question 104 · Multiple choice

Which of the following functions has a vertical asymptote at $x=1$?

1. $y=\dfrac{x+1}{x^2-1}$
2. $y=\dfrac{x-1}{x^2-1}$
3. $y=\dfrac{\sin(x-1)}{x-1}$
4. $y=\ln(x+1)$

Question 105 · Multiple choice

What is $\displaystyle\lim_{x\to\infty}\frac{3x^2-x}{5-2x^2}$?

1. $- \frac{3}{2}$
2. $\frac{3}{2}$
3. $\frac{3}{5}$
4. $-\infty$

Question 106 · Multiple choice

What is $\displaystyle\lim_{x\to-\infty}\frac{2x+1}{\sqrt{x^2+5}}$?

1. $2$
2. $-2$
3. $0$
4. $-\infty$

Question 107 · Multiple choice

What are the horizontal asymptotes of the graph of $y=\dfrac{4e^x+1}{e^x+2}$?

1. $y=4$ only
2. $y=\frac12$ only
3. $y=4$ and $y=\frac12$
4. $y=4$ and $y=-2$

Question 108 · Multiple choice

What is $\displaystyle\lim_{x\to\infty}\left(\sqrt{x^2+6x}-x\right)$?

1. $3$
2. $0$
3. $6$
4. $\infty$

Question 109 · Multiple choice

What is $\displaystyle\lim_{x\to\infty}\frac{5x^3+2}{x^2-7}$?

1. $\infty$
2. $5$
3. $0$
4. $-\infty$

Question 110 · Multiple choice

What is $\displaystyle\lim_{x\to\infty}\frac{x^2+1000x}{e^x}$?

1. $0$
2. $1$
3. $1000$
4. $\infty$

Question 111 · Multiple choice

A function $f$ is continuous on $[1,5]$ with $f(1)=4$ and $f(5)=-2$. For which value of $k$ does the Intermediate Value Theorem guarantee a $c$ in $(1,5)$ with $f(c)=k$?

1. $5$
2. $-3$
3. $0$
4. $\frac{9}{2}$

Question 112 · Multiple choice

The function $g$ is continuous on $[0,6]$; selected values are in the table. What is the least number of solutions of $g(x)=0$ on $[0,6]$ that the table guarantees?

1. $1$
2. $3$
3. $2$
4. $0$

Question 113 · Multiple choice

Let $f(x)=x^3+x-3$. On which interval does the Intermediate Value Theorem guarantee a zero of $f$?

1. $[1,2]$
2. $[0,1]$
3. $[2,3]$
4. $[-1,0]$

Question 114 · Multiple choice

Let $f(x)=\dfrac{1}{x-2}$. Then $f(1)=-1$ and $f(3)=1$, yet $f(x)=0$ has no solution in $[1,3]$. Why does this not contradict the Intermediate Value Theorem?

1. 0 is not between f(1) and f(3)
2. f(1) and f(3) have the same sign
3. the theorem only applies to polynomials
4. f is not continuous on [1, 3]

Question 115 · Multiple choice

Let $f(x)=\cos x-x$. The Intermediate Value Theorem guarantees a zero of $f$ in $(0,1)$ since $f(0)=1$ and $f(1)\approx-0.460$. What is that zero?

1. $0.460$
2. $0.739$
3. $0.540$
4. $0.824$

Question 116 · Free response (9 points)

The graph of the function $f$, defined on $[-3,5]$, is shown. It is made of line segments, with the filled and open circles shown.

1. (a) Find $\displaystyle\lim_{x\to1}f(x)$ and $f(1)$. Is $f$ continuous at $x=1$? Justify your answer. [3 points]
2. (b) Find $\displaystyle\lim_{x\to3^-}f(x)$ and $\displaystyle\lim_{x\to3^+}f(x)$. What type of discontinuity does $f$ have at $x=3$? [2 points]
3. (c) Let $g(x)=f(x)$ for $x\neq1$ and $g(1)=k$. For what value of $k$ is $g$ continuous at $x=1$? Justify your answer. [2 points]
4. (d) Find $\displaystyle\lim_{x\to-1}\frac{f(x)-1}{x+1}$. [2 points]

Question 117 · Free response (9 points)

The temperature of the water in a tank is modeled by a function $W$ that is continuous on $0\le t\le10$, where $W(t)$ is in degrees Fahrenheit and $t$ is in hours. Selected values are given in the table.

1. (a) Find the average rate of change of $W$ over $2\le t\le7$. Indicate units of measure. [2 points]
2. (b) Explain why there must be a time $c$, with $4<c<10$, at which $W(c)=60$. [2 points]
3. (c) Find $\displaystyle\lim_{t\to4}W(t)$. Give a reason. [2 points]
4. (d) After $t=10$ the temperature is modeled by $M(t)=\dfrac{k(t^2-100)}{t-10}$. For what value of $k$ is $\lim_{t\to10^+}M(t)=W(10)$? [3 points]