Prove that the given limit is independent of polynomial $p(x)$.

Question

Prove that, given any polynomial $p(x)$, the value of

$\lim_{x \rightarrow \infty}\dfrac{p(x)}{e^x}$ is independent of polynomial $p(x)$.

The given limit is in $\frac{\infty}{\infty}$ form. If L'Hopitals rule is applied, $\frac{p(x)}{e^x}$ becomes $\infty \cdot 0$. [$\infty$ for $p'(x)$, and $0$ for $\frac{1}{e^\infty}$.

But what is value of $\infty \cdot 0$? I'm sure I must be making a mistake somewhere.

Please help me to solve this problem.

Thank you.

Answer 1

Hint:Use L'Hopital's rule more times to get the numerator zero...

An elegant way is to use induction on the degree of the polynomial and then use L'Hopital's rule to prove that the limit is zero...