The value of lim x→0(|x|)/(x) is

For $x>0$, $\frac{|x|}{x}=1$, while for $x<0$ it equals $-1$. The right-hand limit is $1$ and the left-hand limit is $-1$; they differ, so the limit does not exist. Answering $0$ comes from substituting into the numerator alone. A limit exists only when both one-sided limits agree.