One-sided limits of |x|/x

Right of $0$. If $x>0$ then $|x|=x$, so $f(x)=1$. Hence $\lim_{x\to0^+}f(x)=1$.

Left of $0$. If $x<0$ then $|x|=-x$, so $f(x)=-1$. Hence $\lim_{x\to0^-}f(x)=-1$.

Conclusion. The two one-sided limits exist but are different, so $\lim_{x\to0}f(x)$ does not exist.

Key fact. A limit at a point exists if and only if both one-sided limits exist and are equal. This is the first thing to check when a function has a jump.