Calculus and Its Applications

Marvin L. Bittinger, David J. Ellenbogen, Scott A. Surgent

Chapter 1 Differentiation - all with Video Answers

Educators

Section 1

Limits: A Numerical and Graphical Approach

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Problem 1

Complete each of the following statements.

As $x$ approaches $3,$ the value of $2 x+5$ approaches __________.

Donna Densmore

Numerade Educator

01:40

Problem 2

Complete each of the following statements.

As $x$ approaches $-4,$ the value of $3 x+7$ approaches __________.

Harsh Gadhiya

Numerade Educator

00:33

Problem 3

Complete each of the following statements.

As $x$ approaches __________, the value of $-3 x$ approaches $6.$

Harsh Gadhiya

Numerade Educator

01:05

Problem 4

Complete each of the following statements.

As $x$ approaches ____________, the value of $x-2$ approaches $5.$

Harsh Gadhiya

Numerade Educator

00:20

Problem 5

Complete each of the following statements.

The notation $\lim _{x \rightarrow 4} f(x)$ is read __________.

Harsh Gadhiya

Numerade Educator

00:31

Problem 6

Complete each of the following statements.

The notation $\lim _{x \rightarrow 1} g(x)$ is read ________.

Harsh Gadhiya

Numerade Educator

00:33

Problem 7

Complete each of the following statements.

The notation $\lim _{x \rightarrow 5^{-}} F(x)$ is read ____________.

Harsh Gadhiya

Numerade Educator

00:27

Problem 8

Complete each of the following statements.

The notation $\lim _{x \rightarrow 4^{+}} G(x)$ is read ____________.

Harsh Gadhiya

Numerade Educator

00:25

Problem 9

Complete each of the following statements.

The notation __________ is read "the limit, as $x$ approaches 2 from the right."

Harsh Gadhiya

Numerade Educator

00:25

Problem 10

Complete each of the following statements.

The notation ___________ is read "the limit, as $x$ approaches 3 from the left."

Harsh Gadhiya

Numerade Educator

02:43

Problem 11

When necessary, state that the limit does not exist.
$$\text { Find } \lim _{x \rightarrow 3^{+}} f(x)$$

Harsh Gadhiya

Numerade Educator

00:30

Problem 12

When necessary, state that the limit does not exist.
$$\text { Find } \lim _{x \rightarrow 3^{-}} f(x)$$

Harsh Gadhiya

Numerade Educator

02:03

Problem 13

When necessary, state that the limit does not exist.
$$\text { Find } \lim _{x \rightarrow-1^{-}} f(x)$$

Harsh Gadhiya

Numerade Educator

01:09

Problem 14

When necessary, state that the limit does not exist.
$$\text { Find } \lim _{x \rightarrow-1^{+}} f(x)$$

Harsh Gadhiya

Numerade Educator

00:50

Problem 15

When necessary, state that the limit does not exist.
$$\text { Find } \lim _{x \rightarrow 3} f(x)$$

Harsh Gadhiya

Numerade Educator

00:39

Problem 16

When necessary, state that the limit does not exist.
$$\text { Find } \lim _{x \rightarrow-1} f(x)$$

Harsh Gadhiya

Numerade Educator

01:35

Problem 17

When necessary, state that the limit does not exist.
$$\text { Find } \lim _{x \rightarrow 4} f(x)$$

Harsh Gadhiya

Numerade Educator

01:08

Problem 18

When necessary, state that the limit does not exist.
$$\text { Find } \lim _{x \rightarrow 2} f(x)$$

Harsh Gadhiya

Numerade Educator

03:17

Problem 19

Consider the function g given by
$$g(x)=\left\{\begin{array}{ll}x+6, & \text { for } x<-2 \\ -\frac{1}{2} x+1, & \text { for } x>-2\end{array}\right.$$
If a limit does not exist, state that fact.

$$\lim _{x \rightarrow-2^{-}} g(x)$$

Harsh Gadhiya

Numerade Educator

01:04

Problem 20

Consider the function g given by
$$g(x)=\left\{\begin{array}{ll}x+6, & \text { for } x<-2, \\ -\frac{1}{2} x+1, & \text { for } x>-2.\end{array}\right.$$
If a limit does not exist, state that fact.

$$\lim _{x \rightarrow-2^{+}} g(x)$$

Harsh Gadhiya

Numerade Educator

00:40

Problem 21

Consider the function g given by
$$g(x)=\left\{\begin{array}{ll}x+6, & \text { for } x<-2, \\ -\frac{1}{2} x+1, & \text { for } x>-2.\end{array}\right.$$
If a limit does not exist, state that fact.

$$\lim _{x \rightarrow 4^{+}} g(x)$$

Harsh Gadhiya

Numerade Educator

00:47

Problem 22

Consider the function g given by
$$g(x)=\left\{\begin{array}{ll}x+6, & \text { for } x<-2, \\ -\frac{1}{2} x+1, & \text { for } x>-2.\end{array}\right.$$
If a limit does not exist, state that fact.

$$\lim _{x \rightarrow 4^{-}} g(x)$$

Harsh Gadhiya

Numerade Educator

00:38

Problem 23

Consider the function g given by
$$g(x)=\left\{\begin{array}{ll}x+6, & \text { for } x<-2, \\ -\frac{1}{2} x+1, & \text { for } x>-2.\end{array}\right.$$
If a limit does not exist, state that fact.

$$\lim _{x \rightarrow 4} g(x)$$

Harsh Gadhiya

Numerade Educator

00:27

Problem 24

Consider the function g given by
$$g(x)=\left\{\begin{array}{ll}x+6, & \text { for } x<-2, \\ -\frac{1}{2} x+1, & \text { for } x>-2.\end{array}\right.$$
If a limit does not exist, state that fact.

$$\lim _{x \rightarrow-2} g(x)$$

Harsh Gadhiya

Numerade Educator

00:54

Problem 25

Consider the function g given by
$$g(x)=\left\{\begin{array}{ll}x+6, & \text { for } x<-2, \\ -\frac{1}{2} x+1, & \text { for } x>-2.\end{array}\right.$$
If a limit does not exist, state that fact.

$$\lim _{x \rightarrow 2} g(x)$$

Harsh Gadhiya

Numerade Educator

01:19

Problem 26

Consider the function g given by
$$g(x)=\left\{\begin{array}{ll}x+6, & \text { for } x<-2, \\ -\frac{1}{2} x+1, & \text { for } x>-2.\end{array}\right.$$
If a limit does not exist, state that fact.

$$\lim _{x \rightarrow-4} g(x)$$

Harsh Gadhiya

Numerade Educator

01:47

Problem 27

Use the following graph of $F$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-3} F(x)$$

Harsh Gadhiya

Numerade Educator

01:06

Problem 28

Use the following graph of $F$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 2} F(x)$$

Harsh Gadhiya

Numerade Educator

01:00

Problem 29

Use the following graph of $F$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-2} F(x)$$

Harsh Gadhiya

Numerade Educator

00:39

Problem 30

Use the following graph of $F$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-5} F(x)$$

Harsh Gadhiya

Numerade Educator

01:29

Problem 31

Use the following graph of $F$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 4} F(x)$$

Harsh Gadhiya

Numerade Educator

01:17

Problem 32

Use the following graph of $F$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 6} F(x)$$

Harsh Gadhiya

Numerade Educator

00:40

Problem 33

Use the following graph of $F$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-2^{+}} F(x)$$

Harsh Gadhiya

Numerade Educator

01:10

Problem 34

Use the following graph of $F$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-2^{-}} F(x)$$

Harsh Gadhiya

Numerade Educator

01:34

Problem 35

Use the following graph of $G$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-2} G(x)$$

Harsh Gadhiya

Numerade Educator

00:58

Problem 36

Use the following graph of $G$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 0} G(x)$$

Harsh Gadhiya

Numerade Educator

01:21

Problem 37

Use the following graph of $G$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 1^{-}} G(x)$$

Harsh Gadhiya

Numerade Educator

00:39

Problem 38

Use the following graph of $G$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 1^{+}} G(x)$$

Harsh Gadhiya

Numerade Educator

00:46

Problem 39

Use the following graph of $G$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 1} G(x)$$

Harsh Gadhiya

Numerade Educator

01:37

Problem 40

Use the following graph of $G$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 3^{-}} G(x)$$

Harsh Gadhiya

Numerade Educator

01:37

Problem 41

Use the following graph of $G$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 3^{+}} G(x)$$

Harsh Gadhiya

Numerade Educator

01:37

Problem 42

Use the following graph of $G$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 3} G(x)$$

Harsh Gadhiya

Numerade Educator

01:39

Problem 43

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-3} H(x)$$

Harsh Gadhiya

Numerade Educator

01:03

Problem 44

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-2^{-}} H(x)$$

Harsh Gadhiya

Numerade Educator

01:03

Problem 45

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-2^{+}} H(x)$$

Harsh Gadhiya

Numerade Educator

01:03

Problem 46

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-2} H(x)$$

Harsh Gadhiya

Numerade Educator

02:30

Problem 47

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 1^{-}} H(x)$$

Harsh Gadhiya

Numerade Educator

02:30

Problem 48

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 1^{+}} H(x)$$

Harsh Gadhiya

Numerade Educator

02:30

Problem 49

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 1} H(x)$$

Harsh Gadhiya

Numerade Educator

01:56

Problem 50

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 3^{-}} H(x)$$

Harsh Gadhiya

Numerade Educator

01:56

Problem 51

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.
$$\lim _{x \rightarrow 3^{+}} H(x)$$

Harsh Gadhiya

Numerade Educator

01:56

Problem 52

Use the following graph of $H$ to find each limit. When necessary, state that the limit does not exist.
$$\lim _{x \rightarrow 3} H(x)$$

Harsh Gadhiya

Numerade Educator

01:39

Problem 53

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-1} f(x)$$

Harsh Gadhiya

Numerade Educator

01:52

Problem 54

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 2} f(x)$$

Harsh Gadhiya

Numerade Educator

01:31

Problem 55

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-3} f(x)$$

Harsh Gadhiya

Numerade Educator

01:08

Problem 56

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 0} f(x)$$

Harsh Gadhiya

Numerade Educator

00:59

Problem 57

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow 3} f(x)$$

Harsh Gadhiya

Numerade Educator

01:19

Problem 58

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-4} f(x)$$

Harsh Gadhiya

Numerade Educator

01:19

Problem 59

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-4} f(x)$$

Harsh Gadhiya

Numerade Educator

01:09

Problem 60

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-2} f(x)$$

Harsh Gadhiya

Numerade Educator

00:46

Problem 61

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow \infty} f(x)$$

Harsh Gadhiya

Numerade Educator

00:50

Problem 62

Use the following graph of $f$ to find each limit. When necessary, state that the limit does not exist.

$$\lim _{x \rightarrow-\infty} f(x)$$

Harsh Gadhiya

Numerade Educator

02:01

Problem 63

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$f(x)=|x| ; \quad \text { find } \lim _{x \rightarrow 0} f(x) \text { and } \lim _{x \rightarrow-2} f(x).$$

Harsh Gadhiya

Numerade Educator

01:59

Problem 64

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$f(x)=x^{2} ; \text { find } \lim f(x) \text { and } \lim _{x \rightarrow 0} f(x).$$

Harsh Gadhiya

Numerade Educator

02:12

Problem 65

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$g(x)=x^{2}-5 ; \text { find } \lim _{x \rightarrow 0} g(x) \text { and } \lim _{x \rightarrow-1} g(x).$$

Harsh Gadhiya

Numerade Educator

01:42

Problem 66

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$g(x)=|x|+1 ; \text { find } \lim _{x \rightarrow-3} g(x) \text { and } \lim _{x \rightarrow 0} g(x).$$

Harsh Gadhiya

Numerade Educator

02:27

Problem 67

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$F(x)=\frac{1}{x-3} ; \quad \text { find } \lim _{x \rightarrow 3} F(x) \text { and } \lim _{x \rightarrow 4} F(x).$$

Harsh Gadhiya

Numerade Educator

02:34

Problem 68

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$G(x)=\frac{1}{x+2} ; \text { find } \lim _{x \rightarrow-1} G(x) \text { and } \lim _{x \rightarrow-2} G(x).$$

Harsh Gadhiya

Numerade Educator

01:32

Problem 69

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$f(x)=\frac{1}{x}-2 ; \text { find } \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow 0} f(x).$$

Harsh Gadhiya

Numerade Educator

01:21

Problem 70

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$f(x)=\frac{1}{x}+3 ; \text { find } \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow 0} f(x).$$

Harsh Gadhiya

Numerade Educator

01:21

Problem 71

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$g(x)=\frac{1}{x+2}+4 ; \text { find } \lim _{x \rightarrow \infty} g(x) \text { and } \lim _{x \rightarrow-2} g(x).$$

Harsh Gadhiya

Numerade Educator

01:20

Problem 72

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$g(x)=\frac{1}{x-3}+2 ; \text { find } \lim _{x \rightarrow \infty} g(x) \text { and } \lim _{x \rightarrow 3} g(x).$$

Harsh Gadhiya

Numerade Educator

02:23

Problem 73

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$\begin{aligned}
&F(x)=\left\{\begin{array}{ll}
2 x+1, & \text { for } x<1 \\
x, & \text { for } x \geq 1
\end{array}\right.\\
&\text { Find } \lim _{x \rightarrow 1^{-}} F(x), \lim _{x \rightarrow 1^{+}} F(x), \text { and } \lim _{x \rightarrow 1} F(x).
\end{aligned}$$

Harsh Gadhiya

Numerade Educator

02:58

Problem 74

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$\begin{aligned}
&G(x)=\left\{\begin{array}{ll}
-x+3, & \text { for } x<2 \\
x+1, & \text { for } x \geq 2
\end{array}\right.\\
&\text { Find } \lim _{x \rightarrow 2^{-}} G(x), \lim _{x \rightarrow 2^{+}} G(x), \text { and } \lim _{x \rightarrow 2} G(x).
\end{aligned}$$

Harsh Gadhiya

Numerade Educator

02:07

Problem 75

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$\begin{aligned}
&g(x)=\left\{\begin{array}{ll}
-x+4, & \text { for } x<3 \\
x-3, & \text { for } x>3
\end{array}\right.\\
&\text { Find } \lim _{x \rightarrow 3^{-}} g(x), \lim _{x \rightarrow 3^{+}} g(x), \text { and } \lim _{x \rightarrow 3} g(x).
\end{aligned}$$

Harsh Gadhiya

Numerade Educator

03:11

Problem 76

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$\begin{aligned}
&f(x)=\left\{\begin{array}{ll}
3 x-4, & \text { for } x<1, \\
x-2, & \text { for } x>1.
\end{array}\right.\\
&\text { Find } \lim _{x \rightarrow 1^{-}} f(x), \lim _{x \rightarrow 1^{+}} f(x), \text { and } \lim _{x \rightarrow 1} f(x).
\end{aligned}$$

Harsh Gadhiya

Numerade Educator

03:19

Problem 77

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.
$$G(x)=\left\{\begin{array}{ll}
x^{2}, & \text { for } x<-1, \text { Find } \lim _{x \rightarrow-1} G(x) \\
x+2, & \text { for } x>-1
\end{array}\right.$$

Harsh Gadhiya

Numerade Educator

02:49

Problem 78

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$F(x)=\left\{\begin{array}{ll}-2 x-3, & \text { for } x<-1, \text { Find } \lim _{x \rightarrow-1} F(x). \\x^{3}, & \text { for } x>-1.\end{array}\right.$$

Harsh Gadhiya

Numerade Educator

03:40

Problem 79

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.
$$\begin{aligned}
&H(x)=\left\{\begin{array}{ll}
x+1, & \text { for } x<0 \\
2, & \text { for } 0 \leq x<1 \\
3-x, & \text { for } x \geq 1
\end{array}\right.\\
&\text { Find } \lim _{x \rightarrow 0} H(x) \text { and } \lim _{x \rightarrow 1} H(x).
\end{aligned}$$

Harsh Gadhiya

Numerade Educator

03:06

Problem 80

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.
$$\begin{aligned}
&G(x)=\left\{\begin{array}{ll}
2+x, & \text { for } x \leq-1 \\
x^{2}, & \text { for }-1<x<3 \\
9, & \text { for } x \geq 3
\end{array}\right.\\
&\text { Find } \lim _{x \rightarrow-1} G(x) \text { and } \lim _{x \rightarrow 3} G(x).
\end{aligned}$$

Harsh Gadhiya

Numerade Educator

02:15

Problem 81

In New York City, taxicabs charge passengers $\$ 2.50$ for entering a cab and then $\$ 0.40$ for each one-fifth of $a$ mile (or fraction thereof) traveled. (There are additional charges for slow traffic and idle times, but these are not considered in this problem.) If $x$ represents the distance traveled in miles, then $C(x)$ is the cost of the taxi fare, where

$$\begin{array}{l}
C(x)=\$ 2.50, \quad \text { if } \quad x=0 \\
C(x)=\$ 2.90, \quad \text { if } \quad 0 < x \leq 0.2 \\
C(x)=\$ 3.30, \quad \text { if } \quad 0.2 < x \leq 0.4 \\
C(x)=\$ 3.70, \quad \text { if } \quad 0.4< x \leq 0.6
\end{array}$$

and so on. The graph of $C$ is shown below. (Source: New York City Taxi and Limousine Commission.)

Using the graph of the taxicab fare function, find each of the following limits, if it exists.

$$\lim _{x \rightarrow 0.25^{-}} C(x), \lim _{x \rightarrow 0.25^{+}} C(x), \lim _{x \rightarrow 0.25} C(x)$$

Harsh Gadhiya

Numerade Educator

02:23

Problem 82

and so on. The graph of $C$ is shown below. (Source: New York City Taxi and Limousine Commission.)

Using the graph of the taxicab fare function, find each of the following limits, if it exists.

$$\lim _{x \rightarrow 0.2} C(x), \lim _{x \rightarrow 0.2^{+}} C(x), \lim _{x \rightarrow 0.2} C(x)$$

William Semus

Numerade Educator

01:32

Problem 83

and so on. The graph of $C$ is shown below. (Source: New York City Taxi and Limousine Commission.)

Using the graph of the taxicab fare function, find each of the following limits, if it exists.

$$\lim _{x \rightarrow 0.6^{-}} C(x), \lim _{x \rightarrow 0.6^{+}} C(x), \lim _{x \rightarrow 0.6} C(x)$$

Harsh Gadhiya

Numerade Educator

02:54

Problem 84

The cost of sending a large envelope via U.S. first-class mail is $\$ 0.88$ for the first ounce and $\$ 0.17$ for each additional ounce (or fraction thereof). (Source:
www.usps.com.) If $x$ represents the weight of a large envelope, in ounces, then $p(x)$ is the cost of mailing it, where
$$\begin{array}{l}
p(x)=\$ 0.88, \quad \text { if } \quad 0< x \leq 1, \\
p(x)=\$ 1.05, \quad \text { if } \quad 1 < x \leq 2, \\
p(x)=\$ 1.22, \quad \text { if } \quad 2 < x \leq 3,
\end{array}$$

and so on, up through 13 ounces. The graph of $p$ is shown below

Using the graph of the postage function, find each of the following limits, if it exists.

$$\lim _{x \rightarrow 1^{-}} p(x), \lim _{x \rightarrow 1^{+}} p(x), \lim _{x \rightarrow 1} p(x)$$

Harsh Gadhiya

Numerade Educator

01:07

Problem 85

and so on, up through 13 ounces. The graph of $p$ is shown below

Using the graph of the postage function, find each of the following limits, if it exists.

$$\lim _{x \rightarrow 2^{-}} p(x), \lim _{x \rightarrow 2^{+}} p(x), \lim _{x \rightarrow 2} p(x)$$

Harsh Gadhiya

Numerade Educator

01:16

Problem 86

and so on, up through 13 ounces. The graph of $p$ is shown below

Using the graph of the postage function, find each of the following limits, if it exists.

$$\lim _{x \rightarrow 2.6^{-}} p(x), \lim _{x \rightarrow 2.6^{+}} p(x), \lim _{x \rightarrow 2.6} p(x)$$

Harsh Gadhiya

Numerade Educator

01:06

Problem 87

and so on, up through 13 ounces. The graph of $p$ is shown below

Using the graph of the postage function, find each of the following limits, if it exists.

$$\lim _{x \rightarrow 3} p(x)$$

Harsh Gadhiya

Numerade Educator

01:20

Problem 88

and so on, up through 13 ounces. The graph of $p$ is shown below

Using the graph of the postage function, find each of the following limits, if it exists.

$$\lim _{x \rightarrow 3.4} p(x)$$

Harsh Gadhiya

Numerade Educator

02:00

Problem 89

In $a$ certain habitat, the deer population (in hundreds) as a function of time (in years) is given in the graph of $p$ below.

Use the graph for Exercises.

Find $\lim _{t \rightarrow 1.5^{-}} p(t), \lim _{t \rightarrow 1.5^{+}} p(t),$ and $\lim _{t \rightarrow 1.5} p(t).$

Harsh Gadhiya

Numerade Educator

01:51

Problem 90

In $a$ certain habitat, the deer population (in hundreds) as a function of time (in years) is given in the graph of $p$ below.

Use the graph for Exercises.

Find $\lim _{t \rightarrow 1.75^{-}} p(t), \lim _{t \rightarrow 1.75^{+}} p(t),$ and $\lim _{t \rightarrow 1.75} p(t).$

Harsh Gadhiya

Numerade Educator

01:08

Problem 91

In $a$ certain habitat, the deer population (in hundreds) as a function of time (in years) is given in the graph of $p$ below.

Use the graph for Exercises.

Explain what event(s) might account for the points at which no limit exists.

Harsh Gadhiya

Numerade Educator

01:51

Problem 92

The population of bears in a certain region is given in the graph of $p$ below. Time, $t$, is measured in months.

Use the graph for Exercise.

Find $\lim _{t \rightarrow 0.6^{-}} p(t), \lim _{t \rightarrow 0.6^{+}} p(t),$ and $\lim _{t \rightarrow 0.6} p(t).$

Harsh Gadhiya

Numerade Educator

01:45

Problem 93

The population of bears in a certain region is given in the graph of $p$ below. Time, $t$, is measured in months.

Use the graph for Exercise.

Find $\lim _{t \rightarrow 0.8^{-}} p(t), \lim _{t \rightarrow 0.8^{+}} p(t),$ and $\lim _{t \rightarrow 0.8} p(t).$

Harsh Gadhiya

Numerade Educator

00:49

Problem 94

The population of bears in a certain region is given in the graph of $p$ below. Time, $t$, is measured in months.

Use the graph for Exercise.

Explain what event(s) might account for the points at which no limit exists.

Harsh Gadhiya

Numerade Educator

01:46

Problem 95

Fill in each blank so that $\lim _{x \rightarrow 2} f(x)$ exists.
$$f(x)=\left\{\begin{array}{ll}
\frac{1}{2} x+ & \text { for } x<2 \\
-x+6, & \text { for } x>2
\end{array}\right.$$

Harsh Gadhiya

Numerade Educator

01:29

Problem 96

Fill in each blank so that $\lim _{x \rightarrow 2} f(x)$ exists.
$$f(x)=\left\{\begin{array}{ll}
-\frac{1}{2} x+1, & \text { for } x<2 \\
\frac{3}{2} x+\ldots & \text { for } x>2
\end{array}\right.$$

Harsh Gadhiya

Numerade Educator

01:18

Problem 97

Fill in each blank so that $\lim _{x \rightarrow 2} f(x)$ exists.
$$f(x)=\left\{\begin{array}{ll}
x^{2}-9, & \text { for } x<2 \\
-x^{2}+\ldots, & \text { for } x>2
\end{array}\right.$$

Harsh Gadhiya

Numerade Educator

02:31

Problem 98

Graph the function $f$ given by
$$f(x)=\left\{\begin{array}{ll}
-3, & \text { for } x=-2 \\
x^{2}, & \text { for } x \neq-2
\end{array}\right.$$

Matthew Biollo

Numerade Educator

01:18

Problem 99

Use the GRAPH and TRACE features to find each of the following limits. When necessary, state that the limit does not exist.

a) $\lim _{x \rightarrow-2^{+}} f(x)$
b) $\lim _{x \rightarrow-2^{-}} f(x)$
c) $\lim _{x \rightarrow-2} f(x)$
d) $\lim _{x \rightarrow 2^{+}} f(x)$
e) $\lim _{x \rightarrow 2^{-}} f(x)$
(f) $\operatorname{Does} \lim _{x \rightarrow-2} f(x)=f(-2) ?$
g) $\operatorname{Does} \lim _{x \rightarrow 2} f(x)=f(2) ?$

Use the GRAPH and TRACE features to find each limit. When necessary, state that the limit does not exist.

For $f(x)=\left\{\begin{array}{ll}x^{2}-2, & \text { for } x<0, \\ 2-x^{2}, & \text { for } x \geq 0.\end{array}\right.$
find $\lim _{x \rightarrow 0} f(x)$ and $\lim _{x \rightarrow-2} f(x).$

Matthew Biollo

Numerade Educator

01:32

Problem 100

Use the GRAPH and TRACE features to find each of the following limits. When necessary, state that the limit does not exist.

Use the GRAPH and TRACE features to find each limit. When necessary, state that the limit does not exist.

For $g(x)=\frac{20 x^{2}}{x^{3}+2 x^{2}+5 x},$
find $\lim _{x \rightarrow \infty} g(x)$ and $\lim _{x \rightarrow-\infty} g(x).$

Matthew Biollo

Numerade Educator

01:42

Problem 101

Use the GRAPH and TRACE features to find each of the following limits. When necessary, state that the limit does not exist.

Use the GRAPH and TRACE features to find each limit. When necessary, state that the limit does not exist.

For $f(x)=\frac{1}{x^{2}-4 x-5},$
find $\lim _{x \rightarrow-1} f(x)$ and $\lim _{x \rightarrow 5} f(x).$

Matthew Biollo

Numerade Educator