Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan, Michael Sullivan, lll

Chapter 13

Preview of Calculus: The Limit, Derivative, and Integral - all with Video Answers

Educators

Section 1

Finding Limits Using Tables and Graphs

Problem 1

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
Graph $f(x)=\left\{\begin{array}{ll}3 x-2 & \text { if } x \neq 2 \\ 3 & \text { if } x=2\end{array}\right.$

Amy Jiang

Amy Jiang

Numerade Educator

Problem 2

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
If $f(x)=\left\{\begin{array}{ll}x & \text { if } x \leq 0 \\ 1 & \text { if } x>0\end{array}\right.$ what is $f(0) ?$

Amy Jiang

Numerade Educator

Problem 3

The limit of a function $f(x)$ as $x$ approaches $c$ is denoted by the symbol _______ .

Amy Jiang

Numerade Educator

Problem 4

If a function $f$ has no limit as $x$ approaches $c,$ then we say that $\lim _{x \rightarrow c} f(x)$ _______ _______ _______ .

Amy Jiang

Numerade Educator

Problem 5

$\lim _{x \rightarrow c} f(x)=N$ may be described by saying that the value of $f(x)$ gets closer to $N$ as $x$ gets closer to $c$ but remains unequal to $c$.

Amy Jiang

Numerade Educator

Problem 6

$\lim f(x)$ exists and equals some number for any function $f$ as long as $c$ is in the domain of $f$.

Carson Merrill

Numerade Educator

Problem 7

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 2}\left(4 x^{3}\right)
$$

Amy Jiang

Numerade Educator

Problem 8

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 3}\left(2 x^{2}+1\right)
$$

Amy Jiang

Numerade Educator

Problem 9

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 0} \frac{x+1}{x^{2}+1}
$$

Amy Jiang

Numerade Educator

Problem 10

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 0} \frac{2-x}{x^{2}+4}-
$$

Amy Jiang

Numerade Educator

Problem 11

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 4} \frac{x^{2}-4 x}{x-4}
$$

Amy Jiang

Numerade Educator

Problem 12

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}-3 x}
$$

Amy Jiang

Numerade Educator

Problem 13

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 0}\left(e^{x}+1\right)
$$

Amy Jiang

Numerade Educator

Problem 14

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{2}
$$

Amy Jiang

Numerade Educator

Problem 15

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 0} \frac{\cos x-1}{x}, x \text { in radians }
$$

Amy Jiang

Numerade Educator

Problem 16

Use a table to find the indicated limit.
$$
\lim _{x \rightarrow 0} \frac{\tan x}{x}, x \text { in radians }
$$

Amy Jiang

Numerade Educator

Problem 17

Use the graph shown to determine whether the limit exists. If it does, find its value.
$$
\lim _{x \rightarrow 2} f(x)
$$

Yujie Wang

College of San Mateo

Problem 18

Use the graph shown to determine whether the limit exists. If it does, find its value.
$$
\lim _{x \rightarrow 4} f(x)
$$

James Kiss

Numerade Educator

Problem 19

Use the graph shown to determine whether the limit exists. If it does, find its value.
$$
\lim _{x \rightarrow 2} f(x)
$$

Yujie Wang

College of San Mateo

Problem 20

Use the graph shown to determine whether the limit exists. If it does, find its value.
$$
\lim _{x \rightarrow 2} f(x)
$$

Yujie Wang

College of San Mateo

Problem 21

Use the graph shown to determine whether the limit exists. If it does, find its value.
$$
\lim _{x \rightarrow 3} f(x)
$$

Kim Matthews

Numerade Educator

Problem 22

Use the graph shown to determine whether the limit exists. If it does, find its value.
$$
\lim _{x \rightarrow 4} f(x)
$$

James Kiss

Numerade Educator

Problem 23

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 4} f(x), \quad f(x)=3 x+1
$$

Amy Jiang

Numerade Educator

Problem 24

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow-1} f(x), \quad f(x)=2 x-1
$$

Amy Jiang

Numerade Educator

Problem 25

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 2} f(x), \quad f(x)=1-x^{2}
$$

Amy Jiang

Numerade Educator

Problem 26

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow-1} f(x), \quad f(x)=x^{3}-1
$$

Amy Jiang

Numerade Educator

Problem 27

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow-3} f(x), \quad f(x)=|2 x|
$$

Amy Jiang

Numerade Educator

Problem 28

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 4} f(x), \quad f(x)=3 \sqrt{x}
$$

Amy Jiang

Numerade Educator

Problem 29

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow \pi / 2} f(x), \quad f(x)=\sin x
$$

Amy Jiang

Numerade Educator

Problem 30

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow \pi} f(x), \quad f(x)=\cos x
$$

Amy Jiang

Numerade Educator

Problem 31

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 0} f(x), \quad f(x)=e^{x}
$$

Amy Jiang

Numerade Educator

Problem 32

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 1} f(x), \quad f(x)=\ln x
$$

Amy Jiang

Numerade Educator

Problem 33

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow-1} f(x), \quad f(x)=\frac{1}{x}
$$

Amy Jiang

Numerade Educator

Problem 34

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 2} f(x), \quad f(x)=\frac{1}{x^{2}}
$$

Amy Jiang

Numerade Educator

Problem 35

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 0} f(x), \quad f(x)=\left\{\begin{array}{ll}
x^{2} & \text { if } x \geq 0 \\
2 x & \text { if } x<0
\end{array}\right.
$$

Amy Jiang

Numerade Educator

Problem 36

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 0} f(x), \quad f(x)=\left\{\begin{array}{ll}
x-1 & \text { if } x<0 \\
3 x-1 & \text { if } x \geq 0
\end{array}\right.
$$

Amy Jiang

Numerade Educator

Problem 37

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 1} f(x), \quad f(x)=\left\{\begin{array}{ll}
3 x & \text { if } x \leq 1 \\
x+1 & \text { if } x>1
\end{array}\right.
$$

Amy Jiang

Numerade Educator

Problem 38

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 2} f(x), \quad f(x)=\left\{\begin{array}{ll}
x^{2} & \text { if } x \leq 2 \\
2 x-1 & \text { if } x>2
\end{array}\right.
$$

Amy Jiang

Numerade Educator

Problem 39

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 0} f(x), \quad f(x)=\left\{\begin{array}{ll}
x & \text { if } x<0 \\
1 & \text { if } x=0 \\
3 x & \text { if } x>0
\end{array}\right.
$$

Amy Jiang

Numerade Educator

Problem 40

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 0} f(x), \quad f(x)=\left\{\begin{aligned}
1 & \text { if } x<0 \\
-1 & \text { if } x>0
\end{aligned}\right.
$$

Amy Jiang

Numerade Educator

Problem 41

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 0} f(x), \quad f(x)=\left\{\begin{array}{ll}
\sin x & \text { if } x \leq 0 \\
x^{2} & \text { if } x>0
\end{array}\right.
$$

Amy Jiang

Numerade Educator

Problem 42

Graph each function. Use the graph to find the indicated limit, if it exists.
$$
\lim _{x \rightarrow 0} f(x), \quad f(x)=\left\{\begin{array}{ll}
e^{x} & \text { if } x>0 \\
1-x & \text { if } x \leq 0
\end{array}\right.
$$

Amy Jiang

Numerade Educator

Problem 43

Use a graphing utility to find the indicated limit rounded to two decimal places.
$$
\lim _{x \rightarrow 1} \frac{x^{3}-x^{2}+x-1}{x^{4}-x^{3}+2 x-2}
$$

Amy Jiang

Numerade Educator

Problem 44

Use a graphing utility to find the indicated limit rounded to two decimal places.
$$
\lim _{x \rightarrow-1} \frac{x^{3}+x^{2}+3 x+3}{x^{4}+x^{3}+2 x+2}
$$

Amy Jiang

Numerade Educator

Problem 45

Use a graphing utility to find the indicated limit rounded to two decimal places.
$$
\lim _{x \rightarrow 2} \frac{x^{3}-2 x^{2}+4 x-8}{x^{2}+x-6}
$$

Amy Jiang

Numerade Educator

Problem 46

Use a graphing utility to find the indicated limit rounded to two decimal places.
$$
\lim _{x \rightarrow 1} \frac{x^{3}-x^{2}+3 x-3}{x^{2}+3 x-4}
$$

Amy Jiang

Numerade Educator

Problem 47

Use a graphing utility to find the indicated limit rounded to two decimal places.
$$
\lim _{x \rightarrow-1} \frac{x^{3}+2 x^{2}+x}{x^{4}+x^{3}+2 x+2}
$$

Amy Jiang

Numerade Educator

Problem 48

Use a graphing utility to find the indicated limit rounded to two decimal places.
$$
\lim _{x \rightarrow 3} \frac{x^{3}-3 x^{2}+4 x-12}{x^{4}-3 x^{3}+x-3}
$$

Amy Jiang

Numerade Educator

Problem 49

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Let $A(2,-3)$ and $B(6,-11)$ be points in a plane. Find the distance between the points and the midpoint of the line segment connecting the points.

Harsh Gadhiya

Numerade Educator

Problem 50

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the center, foci, and vertices of the ellipse $\frac{(x-2)^{2}}{9}+\frac{(y+1)^{2}}{13}=1$.

Yujie Wang

College of San Mateo

Problem 51

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Logan invests $$\$ 4000$$ at an annual interest rate of $6 \%$. How much money will she have after 10 years if interest is compounded continuously?

Nicole Hoffman

Numerade Educator

Problem 52

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Assuming $r>0$ and $0 \leq \theta<2 \pi,$ find the polar coordinates of the point whose rectangular coordinates are $(-2,2 \sqrt{3})$.

David Mccaslin

Numerade Educator