Question

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

Answer 1

We have,
$f\left(x\right)=\{\begin{array}{cc}\frac{x}{\left|x\right|},& x\ne 0\\ 0,& x=0\end{array}$
$=\{\begin{array}{c}\frac{x}{-x},x<0\\ 0,x=0\\ \frac{x}{x},x>0\end{array}=\{\begin{array}{c}-1,x<0\\ 0,x=0\\ 1,x>0\end{array}$
Clearly L.H.L $=-1\ne$ R.H.L $=1$
Hence $\underset{x\to 0}{lim}f\left(x\right)$ does not exist