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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.

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  • $\begingroup$ Is $k$ a fixed constant? $\endgroup$
    – Sidharth Ghoshal
    Commented Jun 26, 2023 at 14:05
  • $\begingroup$ If not, can you make explicit what $k(n)$ should be so that your inequalities are still respected $\endgroup$
    – Sidharth Ghoshal
    Commented Jun 26, 2023 at 14:06

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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, $f(n)\to0$.

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