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  "body": "\n## Statement\n\nFor every natural number $n$,\n$$\\lim_{x \\to \\infty} \\frac{e^x}{x^n} = \\infty.$$\n\n## Remarks\n\nFor $x \u003e 0$, the series gives $e^x \u003e \\frac{x^{n+1}}{(n+1)!}$. So $\\frac{e^x}{x^n} \u003e \\frac{x}{(n+1)!}$, which [[tends to infinity|0Y12QYTJK3]].\nEquivalently, $x^n e^{-x} \\to 0$ as $x \\to \\infty$. The same holds for real exponents $n$.\nSo for each polynomial $p$, $p(x) e^{-x} \\to 0$ as $x \\to \\infty$. Mathlib states the main claim as `Real.tendsto_exp_div_pow_atTop`.\n\n## Explanations\n\n### L1 human\n\nThe exponential function eventually grows faster than every power of $x$.\nFor example, $e^x \u003e x^{10}$ for all $x \\ge 36$, but not for $x = 35$.\n",
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