Answer
$\lim\limits_{x\to-2}\frac{2-|x|}{2+x}=1$
Work Step by Step
$A=\lim\limits_{x\to-2}\frac{2-|x|}{2+x}$
We see that $$2-|x|=\left\{
\begin{array} {c l}
2-x && x\geq0\\
2-(-x)=2+x && x<0
\end{array}
\right.$$
In this case, we try to find the limit of the function as $x$ approaches $-2$. Therefore, we only care about the neighbourhood value of $-2$, which is the values very near to $-2$.
In other words, the values of $x\geq0$ are not considered because they are too far from $-2$.
So, $2-|x|=2+x$ as $x\lt0$
Which means,
$A=\lim\limits_{x\to-2}\frac{2+x}{2+x}$
$A=\lim\limits_{x\to-2}1$
$A=1$
