Chapter 11, Problem 6RE

To determine

To graph:

The limit value of the following functions:

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

Explanation of Solution

Given information:

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

Graph:

  

Interpretation:

Know that:

When the limit value of $\underset{x\to a-}{\lim }f\left(x\right)$ and $\underset{x\to a+}{\lim }f\left(x\right)$ is not same then the function does not exist any limit value.

  $\underset{x\to a-}{\lim }f\left(x\right)\ne \underset{x\to a+}{\lim }f\left(x\right)$

From the

The given function $\underset{x\to 2}{\lim }\frac{1}{x-2}$ is graphed as above.

From the graph:

The function reaches $-\infty$ and $\infty$ as it approaches $-2$ from right and left side respectively.

The on sided limit of the function as $x\to -2$ is calculated as:

  $\begin{array}{l}\underset{x\to +2}{\lim }\frac{1}{x-2}=\underset{x\to -2}{\lim }\frac{1}{x-2}\\ \frac{1}{2-2}=\frac{1}{-2-2}\\ \infty \ne -\frac{1}{4}\\ \underset{x\to +2}{\lim }\frac{1}{x-2}\ne \underset{x\to -2}{\lim }\frac{1}{x-2}\end{array}$

Hence, there is no limit value for the function $\underset{x\to 2}{\lim }\frac{1}{x-2}$ .