Question

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

Answer 1

$LHL={lim}_{x\to 0}\left(\frac{1}{x}\right)=-\infty$

$[asx\to 0^{-},\frac{1}{x}\Rightarrow -\infty ]$

$RHL={lim}_{x\to 0^{+}}=\left(\frac{1}{x}\right)=+\infty$

$[asx\to 0^{+},\frac{1}{x}\Rightarrow +\infty ]$

$=LHL\ne RHL$

$\therefore {lim}_{x\to 0}\frac{1}{x}$ does not exist.