# Evaluate $\lim\limits_{x \to 0} f(x)$ where $f(x)=\left\{\begin{array}{1 1}\large\frac{|x|}{x},&x\neq 0\\0,&x=0\end{array}\right.$

The given function is
$f(x)=\left\{\begin{array}{1 1}\large\frac{|x|}{x},&x\neq 0\\0,&x=0\end{array}\right.$
$\lim\limits_{x \to 0^-}f(x) = \lim\limits_{x \to 0^-} \bigg[ \large\frac{|x|}{x} \bigg]$
$= \lim\limits_{x \to 0} = \bigg( \large\frac{-x}{x} \bigg) $$\quad \quad [ when\: x \: is \: negative, |x| = -x] = \lim\limits_{x \to 0} (-1) = -1 \lim\limits_{x \to 0^+}f(x) = \lim\limits_{x \to 0^+} \bigg[ \large\frac{|x|}{x} \bigg] = \lim\limits_{x \to 0} = \bigg( \large\frac{x}{x} \bigg)$$ \quad \quad [ when\: x \: is \: positive, |x| = x]$
$= \lim\limits_{x \to 0} (1)$
$= 1$
It is observed that $\lim\limits_{ x \to 0^-} f(x) \neq \lim\limits_{ x \to 0^+} f(x)$
Hence, $\lim\limits_{ x \to 0} f(x)$ does not exist.