Polynomials versus the exponential

Idea. The exponential eventually beats every power of $x$, however large the power. To prove it, we find a lower bound for $e^x$ that is itself a power of $x$, but of higher degree.

A lower bound. For $x\ge0$, $e^x=\sum_{k=0}^{\infty}\dfrac{x^k}{k!}$ is a sum of non-negative terms, so it is at least any single term. Keeping the term of index $n+1$,
$$e^x\ge\frac{x^{n+1}}{(n+1)!}.$$

Squeeze. For $x>0$,
$$0<\frac{x^n}{e^x}\le\frac{(n+1)!\,x^n}{x^{n+1}}=\frac{(n+1)!}{x}\longrightarrow0.$$
By the squeeze theorem, $\dfrac{x^n}{e^x}\to0$.

Remark. Exponential, powers, logarithms form a hierarchy at $+\infty$: $\ln x\ll x^a\ll e^x$.