Chapter 2.2, Problem 14E

To determine

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

Answer to Problem 14E

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

Explanation of Solution

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

Calculation:

Find the limit graphically.

To graph a function $y=\frac{x}{x-2}$ , 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 $y=\frac{x}{x-2}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{2}\boxed{)}$ .

The display will show the equation,

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

Press the window key and adjust the window to $\left[-2,6\right]$ by $\left[-3,3\right]$ .

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 $f\left(x\right)=\frac{x}{x-2}$ tends to $-\infty$ as $x$ tends to $2$ from the left side.

Now, find the limit by table.

To make the table of function $y=\frac{x}{x-2}$ , 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 $y=\frac{x}{x-2}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{2}\boxed{)}$ .

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 $\boxed{2\text{nd}}\boxed{\text{GRAPH}}$ , and then enter the values of $\text{X}$ as $1.6$ , $1.7$ , $1.8$ , $1.9$ , $1.99$ , $1.999$ and $1.9999$ . The table is shown in window.

  

Figure (2)

As observed from table the value of $\text{Y}$ approaches $-\infty$ as $x$ approaches to $2$ from the left side.

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