Chapter 12.2, Problem 33E

(a)..

To determine

To find: The graphical approximation of the limit by a graphing utility to graph the function.

Answer to Problem 33E

The required limit of the function is $\underset{x\to 1}{\text{lim}}\frac{x-1}{x^2-1}=\frac{1}{2}$ and the graph for it is shown in Figure 4..

Explanation of Solution

Given:

The given function is $\underset{x\to 1}{\text{lim}}\frac{x-1}{x^2-1}$ .

Calculation:

Consider the given equation is,

  $\underset{x\to 1}{\text{lim}}\frac{x-1}{x^2-1}$

Press Y and key in the function the snip for the display is shown in Figure 1

  

Figure 1

Press the window button to choose the proper scale. The display is shown in Figure 2

  

Figure 2

Select the widow button to choose the proper scale as shown in Figure 3

  

Figure 3

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

  

Figure 4

The required limit of the function is,

  $\underset{x\to 1}{\text{lim}}\frac{x-1}{x^2-1}=\frac{1}{2}$

(b)..

To determine

To find: The approximation for the limit of by the table feature of the graphing utility.

Answer to Problem 33E

The value of the limit is limit is $\frac{1}{2}$ .

Explanation of Solution

Calculation:

On the graphing calculator press 2^nd window to choose ask as shown in Figure 5

  

Figure 5

Press 2^nd graph and type the values of x as 0.9, 0.99, 0.999, 1, 1.001, 1.01 and 1.1. The display is shown in Figure 6

  

Figure 6

From the above table the nearest value as x approaches from left of 1 is 0.5.

Hence, the limit is $\frac{1}{2}$ .

(c)..

To determine

To find: The evaluated form of the limit by the appropriate technique.

Answer to Problem 33E

The limit of the function is $\frac{1}{2}$ .

Explanation of Solution

Consider the given expression is,

  $f\left(x\right)=\frac{x-1}{x^2-1}$

Then,

  $\begin{array}{c}f\left(x\right)=\frac{x-1}{\left(x-1\right)\left(x+1\right)}\\ =\frac{1}{2}\end{array}$

Then, the limit of the function is $\frac{1}{2}$ .