Question

Test the continuity of $f\left(x\right)=\{\begin{array}{c}\frac{\left|x\right|}{x},\ne 0\\ 1,x=0\end{array}$ at $x=0$

Answer 1

$f\left(x\right)\{\begin{array}{c}\frac{\left|x\right|}{x},x\ne 0\\ 1,x=0\end{array}$

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

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

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

Hence, $f\left(x\right)$ is Discontinious at $x=0$