I'm not sure but,
I guess (from my book) an asymptotic approximation of $(1-\frac{1}{\sqrt{n}})^n$ would be
$$(1-\frac{1}{\sqrt{n}})^n \space\space \to \space\space e^{-\sqrt{n}-\frac{1}{2} + O(n^{-1/2})}$$ as n increases.
Can you explain to me why?
(That's just my guessing, so you can correct it)