Exponential growth beats polynomial growth

Statement

For every natural number $n$, $$\lim_{x \to \infty} \frac{e^x}{x^n} = \infty.$$

Remarks

For $x > 0$, the series gives $e^x > \frac{x^{n+1}}{(n+1)!}$. So $\frac{e^x}{x^n} > \frac{x}{(n+1)!}$, which tends to infinity. Equivalently, $x^n e^{-x} \to 0$ as $x \to \infty$. The same holds for real exponents $n$. So 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.

Explanations

L1 human

The exponential function eventually grows faster than every power of $x$. For example, $e^x > x^{10}$ for all $x \ge 36$, but not for $x = 35$.