Question

If $f\left(x\right)=\{\begin{array}{c}\frac{x-\left|x\right|}{x},x\ne 0\\ 2,x=0\end{array}$ show that $limf\left(x\right)$ does not exist.

Answer 1

RHL$=\underset{x\to 0^{+}}{lim}\frac{x-|x|}{x}=\underset{x\to 0^{+}}{lim}\frac{x-x}{x}=0$

LHL $=\underset{x\to 0^{-}}{lim}\frac{x-|x|}{x}=\underset{x\to 0^{-}}{lim}\frac{x-(-x)}{x}=2$

since LHL $\ne$ RHL

$\therefore$ limit does not exist.