How to show $\lim_{n\to \infty}\sqrt{n}^n (1 - (1 - 1/(\sqrt{n}^n))^{2^n})/2^n = 1$? How can you show the following?
$$\lim_{n\to \infty}\frac{\sqrt{n}^n \left(1 - \left(1 - \frac{1}{\sqrt{n}^n}\right)^{2^n} \right)}{2^n} = 1$$
It certainly seems to be true numerically when I plot it.

Let $x=\sqrt n^n$, then $2^n=x^2\left(\frac2n\right)^n$. Then we have
$$\displaystyle\begin{align}\lim_{n\to\infty}\frac{x\left(1-\left(1-\frac1x\right)^{x^2\left(\frac2n\right)^n}\right)}{x^2\left(\frac2n\right)^n}=&\lim_{n\to\infty}\frac{1-e^{-x\left(\frac2n\right)^n}}{x\left(\frac2n\right)^n}
=\lim_{n\to\infty}\frac{1-e^{-y}}{y}\end{align}$$
where $y=2^nn^{-\frac n2}\to0$ as $n\to\infty$. Hence the limit is 1.
