Exponential vs Polynomial growth

Question

May somebody give me a prove why $n^{r} < c^{n}$ when $r \in R_{>0}$, $ c \in R_{>1}$ and $ n \in N$ (for all $n$ at some point).

Answer 1

what is, given real $\beta > 0,$ $$ \lim_{x \rightarrow + \infty} \frac{e^{\beta x}}{x} ?$$

It is not necessary to use L'Hospital, it is enough to see that the first derivative is $$ \frac{\left( \beta x - 1 \right) e^{\beta x}}{x^2} $$ and the second is $$ \frac{\left( \beta^2 x^2 - 2 \beta x +2 \right) e^{\beta x}}{x^3}. $$ That is, for $x > 0,$ the second derivative is strictly positive. The original function begins with a vertical asymptote at $x=0,$ descends to a minimum at $x = 1 / \beta,$ but then increases without bound as $x \rightarrow + \infty$

With $\beta = \frac{1}{3},$ here is a nice graph for $x > 0.$