Evaluate \lim\limits_{x \to 0}f(x) where f(x)=\{\dfrac{|x|}{x},x\neq 0,0,x=0

The given function is

 \( \lim\limits_{x \to 0^-}f(x)= \lim\limits_{x \to 0^-}[\dfrac{|x|}{x}]\) \(\)

\( \lim\limits_{x \to 0}(\dfrac{x}{-x})\)

\( \lim\limits_{x \to 0}(-1)\)

= -1

\( \lim\limits_{x \to 0^+}f(x)= \lim\limits_{x \to 0^+}[\dfrac{|x|}{x}]\)

\( \lim\limits_{x \to 0}(\dfrac{x}{x})\)

\( \lim\limits_{x \to 0}(1)\)

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