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

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.

• Sketch a graph of the logarithm of your ratio on the vertical axis and $\sqrt n$ on the horizontal axis. You will meet an old friend. – kimchi lover Feb 3 at 12:11
• The limit is equal to zero – Dr. Sonnhard Graubner Feb 3 at 12:16

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).