Evaluate $lim _{x ightarrow 0} f(x)$, where $f(x)=$

The given function is

$f(x)= \begin{cases}\frac{|x|}{x}, & x \neq 0 \\ 0, & x=0\end{cases}$

$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}\left[\frac{|x|}{x}\right]$

$=\lim _{x \rightarrow 0}\left(\frac{-x}{x}\right) \quad[$ When $x$ is negaitve, $|x|=-x]$

$=\lim _{x \rightarrow 0}(-1)$

$=-1$

$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}}\left[\frac{|x|}{x}\right]$

$=\lim _{x \rightarrow 0}\left[\frac{x}{x}\right] \quad[$ When $x$ is positive, $|x|=x]$

$=\lim _{x \rightarrow 0}(1)$

$=1$

It is observed that $\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)$.

Hence, $\lim _{x \rightarrow 0} f(x)$ does not exist.