Question

Evaluate $li{m}_{x\to 0}f\left(x\right)$, where $f\left(x\right)=\{\begin{array}{cc}\frac{|x|}{x},& x\ne 0\\ 0,& x=0\end{array}$

Answer 1

$LHL={lim}_{x\to 0^{-}}f\left(x\right)$

$\Rightarrow {lim}_{x\to 0^{-}}\frac{|x|}{x}$

as $x\to 0^{-}\Rightarrow x<0$ slightly

$\Rightarrow |x|=-x$

${lim}_{x\to 0^{-}}\frac{-x}{x}=-\mathrm{1....}\left(i\right)$

and

$RHL={lim}_{x\to 0^{+}}f\left(x\right)$

$\Rightarrow {lim}_{x\to 0^{+}}\frac{|x|}{x}$

as $x\to 0^{+}\Rightarrow x>0^{+}$ slightly

$\Rightarrow |x|=x$

${lim}_{x\to 0^{+}}\frac{x}{x}=-1....\left(ii\right)$

So, $LHL\ne RHL$

$\therefore {lim}_{x\to 0}f\left(x\right)$ does not exist.