Question

Evaluate $li{m}_{x\stackrel{0}{\to }}f\left(x\right)$, where $f\left(x\right)=\{\begin{array}{c}\frac{\left|x\right|}{x},x\ne 0\\ 0,x=0\end{array}$

Answer 1

To find the limit at $x=0$ we should have

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

${lim}_{x\to 0^{-}}f\left(x\right)={lim}_{x\to 0^{-}}\frac{\left|x\right|}{x}=\frac{-x}{x}=-1$ since $\left|x\right|=-x$ for $x<0$

${lim}_{x\to 0^{+}}f\left(x\right)={lim}_{x\to 0^{+}}\frac{\left|x\right|}{x}=\frac{x}{x}=1$ since $\left|x\right|=x$ for $x>0$

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

Hence, ${lim}_{x\to 0}f\left(x\right)$ does not exist.