Chapter 2.2, Problem 61E

To determine

Tochoose:The correct option for the value of $\underset{x\to 2^{-}}{\lim }\left(\frac{x}{x-2}\right)$ .

Answer to Problem 61E

The correct choice is (A) such that the value of $\underset{x\to 2^{-}}{\lim }\left(\frac{x}{x-2}\right)$ is $-\infty$ .

Explanation of Solution

Given information:

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

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,

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

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 of function $y=\frac{x}{x-2}$ tends to $-\infty$ as $x$ to $2$ from the left side.

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