Q.Evaluate the one-sided limits and hence $\displaystyle\lim_{x\to 0}\frac{|x|}{x}$, if it exists.

The rule of $|x|$ changes at $0$, so check each side.

By definition $|x|=x$ for $x>0$ and $|x|=-x$ for $x<0$, so the expression must be examined from each side.

Right-hand limit ($x\to0^+$, $x>0$): $\dfrac{|x|}{x}=\dfrac{x}{x}=1$, so $\lim_{x\to0^+}\dfrac{|x|}{x}=1$.

Left-hand limit ($x\to0^-$, $x<0$): $\dfrac{|x|}{x}=\dfrac{-x}{x}=-1$, so $\lim_{x\to0^-}\dfrac{|x|}{x}=-1$.

Since RHL $=1$ and LHL $=-1$ are unequal, the two-sided limit does not exist. …

$\frac{0}{0}$ was avoidable here — the real issue is the sign of $|x|/x$ flipping across $0$. Always …