Question

Does the limit exist? Find the limit, if it exists. If the limit does not exist, explain why?

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

Short answer

The limitof function is $0$.

Step-by-step solution

Step 1. Write the given expression.

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

Step 2. Re-write the given function.

The given is the limit of an absolute value function, so one-sided limits must be used.

The absolute value function can be written as a piecewise function defined over two intervals, one wherein x approaches 0 from the right and one from the left

$f\left(x\right)=\left\{\begin{array}{l}\frac{1}{x}-\left(-\frac{1}{x}\right)\text{if}x<0\\ \left(\frac{1}{x}-\frac{1}{x}\right)\text{if}x\ge 0\end{array}\right.$

Step 3. Find whether the limit of the function exists or not.

We can write the above function as:

$f\left(x\right)=\left\{\begin{array}{l}\frac{2}{x}\text{if}x<0\\ 0\text{if}x\ge 0\end{array}\right.$

Since $f\left(x\right)$$forall$$x\ge 0,\underset{x\to 0^{+}}{\lim }f\left(x\right)=0$

The above definition of the function shows that the limit from the right hand side exists, which is 0 . Therefore, the limit of function is 0 .