recover $\lim\limits_{n\to -\infty}\frac{a^n}{p(n)}=0$ from $\lim\limits_{n\to \infty}\frac{a^n}{p(n)}=\infty$.

Question

I was looking at this post that explains why $\lim\limits_{n\to \infty}\frac{a^n}{p(n)}=\infty$ if $a>1$ and $p(n)$ is a polynomial. Now I was wondering if from this information we could recover that $\lim\limits_{n\to -\infty}\frac{a^n}{p(n)}=0$. So I tried $$\lim\limits_{n\to -\infty}\frac{a^n}{p(n)}=\lim\limits_{n\to -\infty}\frac{(1/a)^{-n}}{q(-n)}=\lim\limits_{n\to \infty}\frac{(1/a)^{n}}{q(n)}$$ but since $1/a<1$ this doesn't really help. Is there a way to recover this?

Answer 1

Sure it helps.

Since $1/a < 1, (1/a)^n \to 0$. Since $|q(n)| \to \infty$, there is no problem.