Question

Show that $\underset{x\to 0}{\lim }\frac{x}{\left|x\right|}$ does not exist.

Answer 1

$\underset{x\to 0}{\lim }\left(\frac{x}{\left|x\right|}\right)$

Left hand limit:
$$\begin{gathered} \underset{x\to 0^{-}}{\lim }\left(\frac{x}{\left|x\right|}\right) \\ \mathrm{Let}x=0-h,\mathrm{where}h\to 0. \\ \Rightarrow \underset{h\to 0}{\lim }\left(\frac{0-h}{\left|0-h\right|}\right) \\ =\underset{h\to 0}{\lim }\left(\frac{-h}{h}\right) \\ =-1 \end{gathered}$$

Right hand limit:
$$\begin{gathered} \underset{x\to 0^{+}}{\lim }\frac{\left(x\right)}{\left|x\right|} \\ \mathrm{Let}x=0+h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }\left(\frac{0+h}{\left|0+h\right|}\right) \\ =\underset{h\to 0}{\lim }\left(\frac{h}{h}\right) \\ =1 \end{gathered}$$

Left hand limit ≠ Right hand limit
$\mathrm{Thus},\underset{x\to 0}{\lim }\left(\frac{x}{\left|x\right|}\right)\mathrm{does}\mathrm{not}\mathrm{exist}.$