Calculate the limit of the following function if it exists
$$\lim\limits_{x o 0} \dfrac{|x|}{x} $$

Question

Calculate the limit of the following function if it exists
$$\lim\limits_{x	o 0}  \dfrac{|x|}{x}  $$

Uniteasy Admin · Accepted Answer

Remove the absolute value sign of the numerator, we get a piecewise function.
$$\begin{cases} -x &	ext{if } x< 0 \ x &	ext{if } x> 0 (x
e 0) \end{cases}$$
So let's calculate the left and right limits separately. 
Left limit,
$$\lim\limits_{x	o 0^-} \dfrac{-x}{x} = -1 $$
Right limit,
$$\lim\limits_{x	o 0^+} \dfrac{x}{x} = 1 $$
 
We get different values. So the limit of the function does not exist since the limit exists if and only if left and right limits exist and are equal.

Question

Answer