# Thread: show limit of e^(1/n) / n = 0

1. ## show limit of e^(1/n) / n = 0

Hi all again,

I know that this limit = 0, but I'm blanking on how to show it..

$\displaystyle \lim_{n\to\infty} \frac{e^{1/n}}{n} = 0$

I'm trying to manipulate it into an acceptable form for l'Hospital, but it isn't coming to me. Thanks for any help!

Brian

2. don't need L'Hopital as directly you get e^(0)/inf = 1/inf = 0

3. Originally Posted by Calculus26
don't need L'Hopital as directly you get e^(0)/inf = 1/inf = 0
Oh! Of course...duh.. heh, thanks.

4. Originally Posted by buckeye1973
Hi all again,

I know that this limit = 0, but I'm blanking on how to show it..

$\displaystyle \lim_{n\to\infty} \frac{e^{1/n}}{n} = 0$

I'm trying to manipulate it into an acceptable form for l'Hospital, but it isn't coming to me. Thanks for any help!

Brian
Originally Posted by Calculus26
don't need L'Hopital as directly you get e^(0)/inf = 1/inf = 0
A little more rigorously note that $\displaystyle \left|\frac{e^{\frac{1}{n}}}{n}\right|\leqslant\fr ac{1}{n}$

5. Originally Posted by Drexel28
A little more rigorously note that $\displaystyle \left|\frac{e^{\frac{1}{n}}}{n}\right|\leqslant\fr ac{1}{n}$

So we will apply the sandwich theorem, right ?

6. Originally Posted by TWiX
So we will apply the sandwich theorem, right ?
True dat.