We have, f (x) = $\begin{cases} \frac{x}{|x|}, & x \neq 0 \\ 0, & x = 0 \end{cases}$ Now, $\lim\limits_{x\to 0^{-}}$ f (x) = $\lim\limits_{x\to 0^{-}}$ $\frac{x}{|x|}$ = $\lim\limits_{x\to 0^{-}}$ $\left(\frac{x}{-x}\right)$ [Since |x| = - x , for x < 0] = $\lim\limits_{x\to 0^{-}}$ ( - 1) = - 1 and $\lim\limits_{x\to 0^{+}}$ f (x) = $\lim\limits_{x\to 0^{+}}$ $\frac{x}{|x|}$ = $\lim\limits_{x\to 0^{+}}$ $\frac{x}{x}$ [Since |x| = x , for x > 0] = $\lim\limits_{x\to 0^{+}}$ (1) = 1 Thus $\lim\limits_{x\to 0^{-}}$ f (x) ≠ $\lim\limits_{x\to 0^{+}}$ f (x) ∴ $\lim\limits_{x\to 0}$ f (x) does not exist.