Exponentials dominate powers in sequences

Statement

Let $k$ be a natural number and let $r$ be a real number with $r > 1$. Then $$\lim_{n \to \infty} \frac{n^k}{r^n} = 0.$$

Remarks

For $k = 1$ and $r = 2$, it says $n / 2^n \to 0$. The terms are $1/2, 2/4, 3/8, 4/16, \dots$. One proof uses the ratio of consecutive terms. The ratio $\frac{(n+1)^k}{n^k} \cdot \frac{1}{r}$ tends to $1/r < 1$, so the terms eventually shrink by a fixed factor. In the language of little o notation, it says $n^k = o(r^n)$. The node exponential growth beats polynomial growth gives the version for real variables.

Explanations

L1 human

A power such as $n^{10}$ grows fast, but an exponential such as $1.01^n$ grows faster in the end. For small $n$ the power can be much larger. For large $n$ the exponential always wins.