Chapter 12.2, Problem 55E

(a)..

To determine

To find: The sketch for the graph of the function for which $f\left(2\right)$ is defined for which the limit of $f\left(x\right)$ as $x$ approaches 2 does not exists.

Answer to Problem 55E

The graph is shown in Figure 1 ..

Explanation of Solution

Consider that the function is defined for the limit but not the function is not defined at the point.

The definition of the limit shows that the limit is defined for only for the value of the function is same as the value of x approaches 2 from the left to the right.

Consider the following equation is,

  $f\left(x\right)=\{\begin{array}{l}\text{4}xx<2\\ x+2x\ge 2\end{array}$

The graph for the above equation is shown in Figure 1

  

Figure 1

The discontinuity in the curve shows that the limit of the function is not defined at the point $x=2$ but the value of the function is defined.

(b)..

To determine

To find: The sketch for the graph of the function for which the limit of $f\left(x\right)$ approaches 1 is 4 but for which $f\left(1\right)\ne 4$ .

Answer to Problem 55E

The limit of the function is $4$ .

Explanation of Solution

Consider the function is,

  $f\left(x\right)=\frac{8\left(x-1\right)}{\left(x^2-1\right)}$

Press Y and key in the function the display is shown in Figure 2

  

Figure 2

Now press the window button and choose the proper scale as shown in Figure 3

  

Figure 3

Press the graph button, the graph is shown in Figure 4

  

Figure 4

To verify if the function satisfy the given equation press the window button and select ask. The display is shown in Figure 5

  

Figure 5

Press the 2^nd graph and type in the value in the X column as shown in Figure 6

  

Figure 6

From the table,

  $\begin{array}{c}\underset{x\to 1}{\lim }=\frac{8\left(x-1\right)}{\left(x^2-1\right),3.998}\\ =4.002\end{array}$

Thus, the limit of the function is $4$ .