Evaluate 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