Chapter 13.2, Problem 34E

To determine

To find the limit it exist, if the limit does not exists, explain why.

Answer to Problem 34E

The solution is 0.

Explanation of Solution

Given information:

Given equation is,

  $\underset{x\to 0^{+}}{\lim }\left(\frac{1}{x}-\frac{1}{\left|x\right|}\right)$

CALCULATION:

To find the limit,

  $\underset{x\to a}{\lim f\left(x\right)}=L$

“The limit of $f\left(x\right)$ as $x$ approaches a, equals $L\text{'}$ .

The limit is obtained as:

  $\begin{array}{l}\underset{x\to 0^{+}}{\lim }\left(\frac{1}{x}-\frac{1}{\left|x\right|}\right)\\ \text{since, }\frac{1}{\left|x\right|}=\left\{\begin{array}{l}\frac{1}{x}\text{ if}x\ge 0\\ -\left(\frac{1}{x}\right)\text{ if }x\le 0\end{array}\right\}\\ =\underset{x\to 0^{-}}{\lim }\left[\frac{1}{x}-\left(\frac{1}{x}\right)\right]\\ =\underset{x\to 0}{\lim }\left(\frac{0}{x}\right)=0\left[\text{taking positive value and simplify}\right]\end{array}$

Hence,

The solution is $\text{'}0\text{'}$