The function $f(x)=\frac{x}{|x|}$ has limit 1 as x approaches 0 from the right, and limit -1 as x approaches 0 from the left. Since these one-sided limit values are NOT the same, there is no single number that f(x) approaches as x approaches 0. So f(x) does not have a two-sided limit at 0.
$$ \lim_{x \rightarrow 0^+} f(x)=1 $$
$$ \lim_{x \rightarrow 0^-} f(x)=-1 $$