How to prove that $e^x$ goes faster to infinity than any polynomial of $x$ without using the Taylor expansion of $e^x$ or L'hopital rule? in other words, the proof that:
$$\lim_{x \to \infty} |x^n|e^{-x}=0$$
I tried to bound the expression from above by a function greater than $|x^n|$ for $x$ greater than some $\delta$ to apply squeeze theorem. I tried proving the limit directly, but both times I could find no excuse for the existence of such $\delta$ without using the known Taylor expansion.
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