If $h(x) = \frac{p(x)}{(x-a)^n},$ where $a$ is a real number, $p(x)$ is a polynomial with $p(a) \ne 0,$ and $n$ is a positive integer, list the conditions under which $\lim_{x\to a}h(x) = -\infty.$
So far, I reasoned that since there is no $(x-a)$ factor in $p(x)$, $n$ must be even, otherwise if it was odd, the limit would be different depending on which side of the limit you take, as the numerator would not be able to make sure the limit is negative on both sides. In addition, the coefficient of the highest degree of $x$ in $p(x)$ has to be negative. I don't think I have the full answer, and a solution outline would be really helpful. If that goes against the spirit of Math.SE, hints would also be appreciated.