Answer
Infinite discontinuity at $-2$. See graph.
Work Step by Step
$f(-2)$ is undefined.
To try and visualize this we use:
$\lim\limits_{x \to -2^{-}} f(x) = \lim\limits_{x \to -2^{-}} \frac{1}{x+2}= -\infty$
$\lim\limits_{x \to -2^{+}} f(x) = \lim\limits_{x \to -2^{+}} \frac{1}{x+2}= \infty$
Both of these answers are not the same, so the $\lim\limits_{x \to -2} f(x)$ does not exist.
