$\lim_{n \to \infty}\frac{e^\sqrt{n}}{c^n}$?

Question

How could I show that the following limit tends to $0$? The constant $c$ can take any value strictly greater than $1.$

$$\lim_{n \to \infty}\dfrac{e^\sqrt{n}}{c^n}$$

I'm having trouble on how to approach this problem.

Answer 1

Your limit can be write as $\lim e^{\sqrt{n}-log(c)n}$, and $\lim (\sqrt{n}-log(c)n)= -\infty$, so $\lim e^{\sqrt{n}-log(c)n}=0$ (or you can make a rigorous proof).