Assume I have the following function


$f\left(n\right)=\frac{-2\sqrt{n}}{A}e^{-A\left(\frac{k}{\sqrt{n}}\right)}\left(e^{-\frac{A}{\sqrt{n}}}-1\right)$


Where $n-k\gg1$ and $k\gg\sqrt{n}$ and $A=2$ is a constant.


And $n\rightarrow\infty$.


My thoughts is that $f\left(n\right)\rightarrow 0$ (I have ran some examples in walfram alpha and desmos)
But I am not quite sure how to approach this problem and I would like some help.



asymptotic behavior of function $f\left(n\right)=\frac{-2\sqrt{n}}{A}e^{-A\left(\frac{k}{\sqrt{n}}\right)}\left(e^{-\frac{A}{\sqrt{n}}}-1\right)$
You have
$$f(n)=\sqrt{n} (1-e^{-2/\sqrt{n}})
e^{-2k/\sqrt{n}}\sim2e^{-2k/\sqrt{n}}\to0,
$$
since $k>>\sqrt{n}$. So, $f(n)\to0$.


