Question

f(x)=$\{\begin{array}{c}\frac{|x|}{x},ifx\ne 0\\ 0,ifx=0\end{array}$

Answer 1

Here, f(x)=$\{\begin{array}{c}\frac{|x|}{x},ifx\ne 0\\ 0,ifx=0\end{array}$

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

Putting x=0-h as $x\to 0^{-}$ when $x\to 0^{-}$

$\therefore$ $\underset{h\to 0}{lim}$ $\frac{|0-h|}{o-h}$ $\underset{h\to 0}{lim}$ $\frac{h}{-h}=-1$ [$\therefore$|-h1=h]

RHL = $\underset{x\to 0^{+}}{lim}$ $f(\times )$ =$\underset{x\to 0^{+}}{lim}$ $\frac{|\times |}{x}$

Putting x=0 +h $x\to 0^{+}$ ; $h\to 0$

$\underset{h\to 0}{lim}$ $\frac{|0-h|}{o+h}$ $\underset{h\to 0}{lim}$ $\frac{h}{h}=1$

$\therefore$ LHL $\text{/}$ = RHL. Thus, $f(\times )$ is discontinuous at x=0.