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Will the limit as x approaches infinity of a polynomial over an exponential ever tend to infinity? I know the limit as x approaches infinity of x^2/e^x tends to 0, but will we ever be able to generate a power to the x large enough that it dominates over the exponential and doesn't tend to 0?

raveen
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1 Answers1

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Never. For any polynomial $P$ of degree $n$, apply L'hospital's rule $n$ times to $\lim_{x\to \infty}\frac{P(x)}{e^x}$

Adam Martens
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  • I understand we can apply L'hospital's rule and the polynomial will eventually end up a constant. But, even if we used the largest number known to man in the degree of the polynomial, it would still tend to 0? Would it not be possible to generate a number that essentially dominates the infinity, or not even dominate but reach a limit >0? – raveen Dec 05 '19 at 00:32
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    @raveen fix any $n\in\mathbb{N}$ to be the degree of the polynomial $P$. You can take $n$ as big as you want. When you do $P^{(n+1)},$ you get constant $0$, and $e^{x}$ remains $e^{x}$. – Mateus Rocha Dec 05 '19 at 00:48