Answer
Discontinuous at $ x=0$
Work Step by Step
Recall that if $ f $ is a polynomial function, then we have $\lim_\limits{x\to a}f(x)=f(a)$.
$\lim_\limits{x \to 0^{-}} f(x)= \lim_\limits{x \to 0^{+}}\dfrac{x^2+5x}{x^2-5x}=-1$
and $\lim_\limits{x \to 0^{+}} f(x)= \lim_\limits{x \to 0^{+}}\dfrac{x^2+5x}{x^2-5x}=-1$
So, $\lim_\limits{x \to 0^{-}} f(x)= \lim_\limits{x \to 0^{+}}f(x)$ exists.
Now, $\lim_\limits{x \to 0^{-}} f(x)=-1$
and $ f(0)=-2$
so, $\lim_\limits{x \to 0} f(x) \neq f(0)$
Therefore, the function is Discontinuous at $ x=0$
