Limits involving absolute values often involve breaking things into cases.
Remember that
$$|f(x)|=\begin{cases}
f(x), &\text{ if } f(x)\ge0;\\
-f(x), &\text{ if } f(x)\le0.\\
\end{cases}$$
By studying these cases separately,
we can often get a good picture of what a function is doing just to the left
of $x=a$, and just to the right of $x=a$. By combining these, we can understand
the limit as $x \to a$.