Question

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

Answer 1

$f\left(x\right)=\{\begin{array}{cc}\frac{\left|x\right|}{x},& x\ne 0\\ 0,& x=0\end{array}$
$\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0}{lim}\left(\frac{-x}{x}\right)=-1$
$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0}{lim}\left(\frac{x}{x}\right)=1$
Clearly $\underset{x\to 0^{-}}{lim}f\left(x\right)\ne \underset{x\to 0^{+}}{lim}f\left(x\right)\cdot$
Hence $\underset{x\to 0}{lim}f\left(x\right)$ does not exist.