Chapter 2.1, Problem 35E

To determine

To show: The limit $\underset{x\to 1}{\lim }\left(\frac{x^2-4}{x-1}\right)$ does not exist with the help of graph.

Explanation of Solution

Given information: The limit is $\underset{x\to 1}{\lim }\left(\frac{x^2-4}{x-1}\right)$ .

Proof:

Let $y=\frac{x^2-4}{x-1}$ to find the value of $\underset{x\to 1}{\lim }\left(\frac{x^2-4}{x-1}\right)$ graphically.

To graph a function $y=\frac{x^2-4}{x-1}$ , 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 $y=\frac{x^2-4}{x-1}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{-}\boxed{4}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{1}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(\text{X}^2-4\right)/\left(\text{X}-1\right)$

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)

The left hand limit $\underset{x\to 1^{-}}{\lim }\left(\frac{x^2-4}{x-1}\right)$ is a positive number and the right hand limit $\underset{x\to 1^{+}}{\lim }\left(\frac{x^2-4}{x-1}\right)$ is a negative number. So, the left hand limit is not equal to the right hand limit.

Hence, it is proved that the limit $\underset{x\to 1}{\lim }\left(\frac{x^2-4}{x-1}\right)$ does not exist.