Question

Show that $\underset{x\to 0}{\lim }\frac{1}{x}$ does not exist.

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left(\frac{1}{x}\right) \\ \mathrm{Left}\mathrm{hand}\mathrm{limit}: \\ \underset{x\to 0^{-}}{\lim }\left(\frac{1}{x}\right) \\ \mathrm{Let}x=0-h,\mathrm{where}h\to 0 \\ \underset{h\to 0}{\lim }\left(\frac{1}{0-h}\right)=-\infty \\ \mathrm{Right}\mathrm{hand}\mathrm{limit}: \\ \underset{x\to 0^{+}}{\lim }\left(\frac{1}{x}\right) \\ \mathrm{Let}x=0+h,\mathrm{where}h\to 0 \\ \underset{h\to 0}{\lim }\left(\frac{1}{0+h}\right)=\infty \\ \underset{x\to 0^{-}}{\lim }\left(\frac{1}{x}\right)\ne \underset{x\to 0^{+}}{\lim }\left(\frac{1}{x}\right) \\ \mathrm{Thus},\underset{x\to 0}{\lim }\left(\frac{1}{x}\right)\mathrm{does}\mathrm{not}\mathrm{exist}. \end{gathered}$$