Application of the Taylor series.

Question

I am trying to look for the following limit, $\lim_{x\to \infty} \frac{x^3}{e^x}$. I have tried using the L'Hôpital's rule and I get $0$ as a result. However, I am curious to know how we can get the same result using the taylor series as $e^x =\sum_{n=0}^\infty \frac{x^n}{n!}$.I am stuck and would like some help. Thank you in advance.

Answer 1

For $x>0$, $e^x=1+x+\cdots+x^4/24+\cdots>x^4/24$. Therefore $x^3/e^x<24/x$.