We have, f (x) = $\begin{cases} \frac{|x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$ Now , $\lim\limits_{x\rightarrow0^{-}}$ f (x) = $\lim\limits_{x\rightarrow0^{-}}$ $\frac{|x|}{x}$ = $\lim\limits_{x\rightarrow0^{-}}$ $\left(-\frac{x}{x}\right)$ [Since |x| = - x for x < 0] = $\lim\limits_{x\rightarrow0^{-}}$ (- 1) = - 1 and $\lim\limits_{x\rightarrow0^{+}}$ f (x) = $\lim\limits_{x\rightarrow0^{+}}$ $\frac{|x|}{x}$ = $\lim\limits_{x\rightarrow0^{+}}$ $\left(\frac{x}{x}\right)$ [Since |x| = x , for x > 0] = $\lim\limits_{x\rightarrow0^{+}}$ (1) = 1 Since $\lim\limits_{x\rightarrow0^{-}}$ f (x) ≠ $\lim\limits_{x\rightarrow0^{+}}$ f (x) Thus $\lim\limits_{x\rightarrow0}$ f (x) does not exist.