Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 42

Determine the limit $\lim\limits_{x \rightarrow -2} \displaystyle \frac{2-|x|}{2+x}$, if it exists. If the limit does not exist, explain why.

The function contains an absolute value, therefore, we evaluate its left and right hand limit


$
\begin{equation}
\begin{aligned}
\text{For the right hand limit}\\
\lim\limits_{x \to -2^+} \frac{2-|x|}{2+x} & = \lim\limits_{x \to -2^+} \frac{2-x}{2+x}\\
\phantom{x} & = \lim\limits_{x \to -2^+} \frac{2-x}{2+x}\\
\phantom{x} & = \infty\\
\end{aligned}
\end{equation}
$




$
\begin{equation}
\begin{aligned}
\text{For the left hand limit}\\
\lim\limits_{x \to -2^-} \frac{2-|x|}{2+x} & = \lim\limits_{x \to -2^-} \frac{2-(-x)}{2+x)}\\
\phantom{x} & = \lim\limits_{x \to -2^-} \frac{\cancel{2+x}}{\cancel{2+x}}\\
\phantom{x} & = \lim\limits_{x \to -2^-} 1\\
\phantom{x} & = 1
\end{aligned}
\end{equation}
$


The left and right hand limits are different. Therefore, the limit does not exist