Chapter 2.2, Problem 13E

To determine

To find: The value of $\underset{x\to 2^{+}}{\lim }\left(\frac{1}{x-2}\right)$ with the help of graph and table.

Answer to Problem 13E

The value of $\underset{x\to 2^{+}}{\lim }\left(\frac{1}{x-2}\right)$ is $\infty$ .

Explanation of Solution

Given information: The limit is $\underset{x\to 2^{+}}{\lim }\left(\frac{1}{x-2}\right)$ .

Calculation:

Find the limit graphically.

To graph a function $y=\frac{1}{x-2}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{1}{x-2}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{2}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=1/\left(\text{X}-2\right)$

Press the window key and adjust the window to $\left[-2,6\right]$ by $\left[-1,5\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

As observed from graph, the function $f\left(x\right)=\frac{1}{x-2}$ tends to $\infty$ as $x$ tends to $2$ from the right side.

Now, find the limit by table.

To make the table of function $y=\frac{1}{x-2}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{1}{x-2}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{2}\boxed{)}$ .

First set the Table setup, Enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{WINDOW}}$ , Then set to independent variable to ask.

To draw the table, enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{GRAPH}}$ , the enter the values of $\text{X}$ as $2.4$ , $2.3$ , $2.2$ , $2.1$ , $2.01$ , $2.001$ and $2.0001$ . The table is shown in window.

  

Figure (2)

As observed from table the value of $\text{Y}$ increases very rapidly to the positive side as $x$ approaches to $2$ from the right side.

Therefore, the value of $\underset{x\to 2^{+}}{\lim }\left(\frac{1}{x-2}\right)$ is $\infty$ .