Question

Check the continuity of the function $f\left(x\right)=\{\begin{array}{cc}\frac{|x|}{x},& x\ne 0\\ 0,& x=0\end{array}\text{at}x=0$

Answer 1

$f\left(x\right)=\{\begin{array}{cc}\frac{|x|}{x},& x\ne 0\\ 0,& x=0\end{array}\text{at}x=0$

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}\frac{|x|}{x}=\underset{x\to 0^{+}}{lim}\frac{x}{x}=1$

$\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}\frac{|x|}{x}=\underset{x\to 0^{-}}{lim}\frac{-x}{x}=-1$

Here $\underset{x\to 0^{+}}{lim}f\left(x\right)\ne \underset{x\to 0^{-}}{lim}f\left(x\right)\ne f\left(0\right)$

So $f\left(x\right)$ is not continuous at $x=0$