## Lim of An=n^2*exp(-sqrt(n))

Hi all, my problem regards this limit:

$$\lim_{n\to\infty}n^2e^{(-\sqrt{n})}$$

Obviously equals 0, but I can't find how to show it.
Tried the squeeze theorem (coudn't find any propriate upper bound)
Ratio test won't seem to work..
I do realize the reason for that is that the set approaches 0 starting at heigher n's..

Anyway.. how can I prove convergence and find the limit in a formal way? thanks!

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 Recognitions: Science Advisor Simple method: Let m=√n, so the problem is limit m -> ∞ m4/em. em = 1 + m + m2/2! + m3/3! + m4/4! + m5/5! + .... It is obvious from the 5th term on the denominator of the fraction swamps the numerator.
 I've tried changing variables like you did and got m4/em, which does seem nicer.. But is using taylor expansion the only way to solve here? I'm pretty sure that's not what the course staff expected us to do..

Recognitions:
Science Advisor

## Lim of An=n^2*exp(-sqrt(n))

Have learned L'Hopital's rule?
If so, use that. Take 5 derivatives of the numerator and the denominator and get 0/em.

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