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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.

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1 Answer 1

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

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