Chapter 12.1, Problem 67E

(a)

To determine

To find: The graph for the corresponding functions by the use of standard viewing window. Also, determine if the graph reveal whether it is possible to reach the limit.

Answer to Problem 67E

The graph for the function is defined by the use of decimal setting and what the graph reveal and if it reach the limit is described.

Explanation of Solution

Given:

The given limit are,

  $\begin{array}{l}\underset{x\to 2}{\text{lim}}\left(5x+4\right)\\ \underset{x\to 1}{\text{lim}}\left(2x^2+x-4\right)\\ \underset{x\to -1}{\text{lim}}\left(\frac{x+1}{x^2-x+2}\right)\\ \underset{x\to 0}{\text{lim}}\left(\frac{\tan x}{2x}\right)\end{array}$

Also,

  $\underset{x\to 0}{\text{lim}}\left(\frac{\tan x}{2x}\right)$

Calculation

Consider the function $f\left(x\right)=5x+4$ .

Press Y= and key in the function as shown in Figure 1

  

Figure 1

Press the graph button to trace the function as shown in Figure 2

  

Figure 2

The graph of the function is defined at the required point the limit of the function exists at the defined point.

Consider the function $f\left(x\right)=2x^2+x-4$ .

Press Y= and key in the function as shown in Figure 3

  

Figure 3

Press the graph button to trace the function as shown in Figure 4

  

Figure 4

Consider the function $\frac{x-3}{x^2-9}$ .

Press Y= and key in the function as shown in Figure 5

  

Figure 5

Press the graph button to trace the function as shown in Figure 6

  

Figure 6

The graph of the function is defined at the required point the limit of the function exists at the defined point.

Consider the function $f\left(x\right)=\frac{x+1}{x^2-x-2}$ .

Press Y= and key in the function as shown in Figure 7

  

Figure 7

Press the graph button to trace the function as shown in Figure 8

  

Figure 8

The graph of the function is defined at the required point the limit of the function exists at the defined point.

Consider the function $f\left(x\right)=\frac{\sin 2x}{x}$ .

Press Y= and key in the function as shown in Figure 9

  

Figure 9

Press the graph button to trace the function as shown in Figure 10

  

Figure 10

The graph of the function is defined at the required point the limit of the function exists at the defined point.

Consider the function $f\left(x\right)=\frac{\tan x}{2x}$ .

Press Y= and key in the function as shown in Figure 11

  

Figure 11

Press the graph button to trace the function as shown in Figure 12

  

Figure 12

The graph of the function is defined at the required point the limit of the function exists at the defined point.

(b)

To determine

To find: The graph for the corresponding functions by the use of decimal settings window. Also, determine if the graph reveal whether it is possible to reach the limit.

Answer to Problem 67E

The graph for the function is defined by the use of decimal setting and what the graph reveal and if it reach the limit is described.

Explanation of Solution

Given:

The given limits are,

  $\begin{array}{l}\underset{x\to 2}{\text{lim}}\left(5x+4\right)\\ \underset{x\to 1}{\text{lim}}\left(2x^2+x-4\right)\\ \underset{x\to -1}{\text{lim}}\left(\frac{x+1}{x^2-x+2}\right)\\ \underset{x\to 0}{\text{lim}}\left(\frac{\tan x}{2x}\right)\end{array}$

Also,

  $\underset{x\to 0}{\text{lim}}\left(\frac{\tan x}{2x}\right)$

Calculation

In order to change the window setting to decimal setting of the individual graphs proceed as follows.

Consider the function $f\left(x\right)=5x+4$ .

Press Y= and key in the function as shown in Figure 13

  

Figure 13

Press the windows to change the settings to decimal settings as shown in Figure 14

  

Figure

14

Press the graph button to trace the function as shown in Figure 15

  

Figure 15

The graph of the function is defined at the required point the limit of the function exists at the defined point.

Consider the function $f\left(x\right)=2x^2+x-4$ .

Press Y= and key in the function as shown in Figure 3

  

Figure 16

Press the windows to change the settings to decimal settings as shown in Figure 17

  

Figure 17

Press the graph button to trace the function as shown in Figure 18

  

Figure 18

The graph of the function is defined at the required point the limit of the function exists at the defined point.

Consider the function $\frac{x-3}{x^2-9}$ .

Press Y= and key in the function as shown in Figure 19

  

Figure 19

Press the windows to change the settings to decimal settings as shown in Figure 20

  

Figure 20

Press the graph button to trace the function as shown in Figure 21

  

Figure 21

The graph of the function is defined at the required point the limit of the function exists at the defined point.

Consider the function $f\left(x\right)=\frac{x+1}{x^2-x-2}$ .

Press Y= and key in the function as shown in Figure 22

  

Figure 22

Press the windows to change the settings to decimal settings as shown in Figure 23

  

Figure 23

Press the graph button to trace the function as shown in Figure 24

  

Figure 24

The graph of the function is defined at the required point the limit of the function exists at the defined point.

Consider the function $f\left(x\right)=\frac{\sin 2x}{x}$ .

Press Y= and key in the function as shown in Figure 25

  

Figure 25

Press the windows to change the settings to decimal settings as shown in Figure 26

  

Figure 26

Press the graph button to trace the function as shown in Figure 27

  

Figure 27

The graph of the function is defined at the required point the limit of the function exists at the defined point.

Consider the function $f\left(x\right)=\frac{\tan x}{2x}$ .

Press Y= and key in the function as shown in Figure 28

  

Figure 28

Press the windows to change the settings to decimal settings as shown in Figure 29

  

Figure 29

Press the graph button to trace the function as shown in Figure 30

  

Figure 30

The graph of the function is defined at the required point the limit of the function exists at the defined point.