Answer
Discontinuous at $ x=-2$
Work Step by Step
Recall that if $ f $ is a polynomial function, then we have $\lim_\limits{x\to a}f(x)=f(a)$.
$\lim_\limits{x \to -2} f(x)= \lim_\limits{x \to -2} \dfrac{x^2-4}{x+2}= \lim_\limits{x \to -2} \dfrac{(x-2)(x+2)}{x+2}=-4$
and $\lim_\limits{x \to -2} f(x)= \lim_\limits{x \to -2} \dfrac{x^2-4}{x+2}= \lim_\limits{x \to -2} \dfrac{(x-2)(x+2)}{x+2}=-4$
So, $\lim_\limits{x \to 2} f(x)= \lim_\limits{x \to -2}f(x)$ exists.
Now, $\lim_\limits{x \to -2} f(x)=-2$
so, $\lim_\limits{x \to -2} f(x) \ne -4$
Therefore, the function is discontinuous at $ x=-2$
