Question

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

Answer 1

$$\begin{gathered} f\left(x\right)=\left\{\begin{array}{ll}\frac{\left|x\right|}{x},& x\ne 0\\ 0,& x=0\end{array}\right. \\ \mathrm{LHL}: \\ \underset{x\to 0^{-}}{\lim }\left(\frac{\left|x\right|}{x}\right) \\ \mathrm{Let}x=0-h,\mathrm{where}h\to 0. \\ \Rightarrow \underset{h\to 0}{\lim }\left(\frac{\left|0-h\right|}{-h}\right) \\ =\underset{h\to 0}{\lim }\left(\frac{h}{-h}\right)=-1 \\ \mathrm{RHL}: \\ \underset{x\to 0^{+}}{\lim }f\left(x\right) \\ =\underset{x\to 0^{+}}{\lim }\left(\frac{\left|x\right|}{x}\right) \\ \mathrm{Let}x=0+h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }\left(\frac{\left|0+h\right|}{0+h}\right)=1 \\ \mathrm{LHL}\ne \mathrm{RHL} \\ \underset{x\to 0}{\mathrm{Thus},\lim }f\left(x\right)\mathrm{does}\mathrm{not}\mathrm{exist}. \end{gathered}$$