Question

What is the limit as x is approaching 0 of the absolute value of x

Answer 1

Recall that, an absolute value expression is a piece-wise in effect, thus we'd check the single-sided limits,

$\bf \lim\limits_{x\to 0}~|x|\implies \begin{cases} \lim\limits_{x\to 0^-}~-x\\\\ \lim\limits_{x\to 0^+}~+x \end{cases}\implies \begin{cases} 0\\\\ 0 \end{cases}\implies \lim\limits_{x\to 0}~|x|=0$