Question

Show that does not exist.

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right) \\ \mathrm{L}.\mathrm{H}.\mathrm{L} \\ \underset{x\to 0^{-}}{\lim }f\left(x\right) \\ \underset{x\to 0^{-}}{\lim }\sin \left(\frac{1}{x}\right) \\ \mathrm{Let}x=0-h\mathrm{where}h\to 0 \\ =\underset{h\to 0}{\lim }\sin \left(-\frac{1}{h}\right) \\ =-\underset{h\to 0}{\lim }\sin \left(\frac{1}{h}\right) \end{gathered}$$

$\underset{x\to 0^{-}}{\lim }f\left(x\right)$ = – (An oscillating number that oscillates between –1 and 1)

$$\begin{gathered} \mathrm{Thus},\underset{x\to 0^{-}}{\lim }f\left(x\right)\mathrm{does}\mathrm{not}\mathrm{exist}. \\ \mathrm{Similarly},\underset{x\to 0^{+}}{\lim }f\left(x\right)\mathrm{does}\mathrm{not}\mathrm{exist}. \end{gathered}$$