I'd would like to know how to get the answer of the following problem:
$\lim_{n \to \infty} \left(2\sqrt{n}\left(\sqrt{n+1}-\sqrt{n}\right)\right)^n$
I know that the answer is $\frac{1}{e^{1/4}}$, but I can't figure out how to get there. This is a homework for my analysis class, but I can't solve it with any of the tricks we learned there.
This is what I got after a few steps, however it feels like this is a dead end:
$\lim_{n \to \infty} \left(2\sqrt{n}\times \frac{\left(\sqrt{n+1}-\sqrt{n}\right)\times \left(\sqrt{n+1}+\sqrt{n}\right)}{\left(\sqrt{n+1}+\sqrt{n}\right)}\right)^n=$ $\lim_{n \to \infty} \left(\frac{2\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}\right)^n$
Thanks for your help in advance.