Note that $\displaystyle \left|x\right|=\left\{\begin{array}{rl} x, & \text{ if } x\geq0\\ {\color{red}-x}, & \text{ if } x<0\end{array}\right.$
So $\displaystyle \lim_{x\to-2}\frac{2-\left|x\right|}{2+x}=\lim_{x\to-2}\frac{2-(-x)}{2+x}=\lim_{x\to-2}\frac{2+x}{2+x}=1$
Does this make sense?