Answer
The function $f(x)$ is not continuous at $x=-2$.
Work Step by Step
$f(x)=\dfrac{x^2-4}{x+2}$
We need to check whether the function $f(x)$ is continuous at $x=-2$.
We check the left-hand and right-hand limits. If they are equal to each other and the value of the function at the limit, then the function is continuous at that point.
Since, $f(-2)=4$
$\lim\limits_{x \to -2}f(x)=\lim\limits_{x \to -2}\dfrac{x^2-4}{x+2} \\=\lim\limits_{x \to -2}\dfrac{(x+2)(x-2)}{x+2} \\=\lim\limits_{x \to -2}(x-2)\\=(-2)-2\\=-4$
We can see that the function $f(x)$ at $x=-2$ is $4 \ne -4$, so the function cannot be defined at $x=-2$.
Therefore, the function $f(x)$ is not continuous at $x=-2$.
