recall that, an absolute value expression is a piece-wise in effect, thus we'd check the single-sided limits,
[tex]\bf \lim\limits_{x\to 0}~|x|\implies
\begin{cases}
\lim\limits_{x\to 0^-}~-x\\\\
\lim\limits_{x\to 0^+}~+x
\end{cases}\implies
\begin{cases}
0\\\\
0
\end{cases}\implies \lim\limits_{x\to 0}~|x|=0[/tex]
