Chapter 2.1, Problem 23E

To determine

To check: Whether the direct substitution can be used for the limit $\underset{x\to 0}{\lim }\frac{\left|x\right|}{x}$ or not.

Answer to Problem 23E

The direct substitution cannot be used for the limit $\underset{x\to 0}{\lim }\frac{\left|x\right|}{x}$ as the function $\frac{\left|x\right|}{x}$ is not defined at $x=0$ . The limit $\underset{x\to 0}{\lim }\frac{\left|x\right|}{x}$ does not exist.

Explanation of Solution

Given information: The limit is $\underset{x\to 0}{\lim }\frac{\left|x\right|}{x}$ .

Calculation:

Substitute $0$ for $x$ in the function.

  $\underset{x\to 0}{\lim }\frac{\left|x\right|}{x}=\frac{\left|0\right|}{0}$

The limit is not defined.

Calculate the left hand limit of the function at $x=0$ .

  $\begin{array}{c}\underset{x\to 0^{-}}{\lim }\frac{\left|x\right|}{x}=\underset{x\to 0^{-}}{\lim }\frac{-x}{x}\\ =\underset{x\to 0^{-}}{\lim }\left(-1\right)\\ =-1\end{array}$

Now, calculate the right hand limit of the function at $x=0$ .

  $\begin{array}{c}\underset{x\to 0^{+}}{\lim }\frac{\left|x\right|}{x}=\underset{x\to 0^{+}}{\lim }\frac{x}{x}\\ =\underset{x\to 0^{+}}{\lim }\left(1\right)\\ =1\end{array}$

As the left hand limit is not equal to the right hand limit of the given function at $x=0$ . So, the limit $\underset{x\to 0}{\lim }\frac{\left|x\right|}{x}$ does not exist.

Therefore, the limit $\underset{x\to 0}{\lim }\frac{\left|x\right|}{x}$ does not exist and direct substitution cannot be used.