# limit involving exponential

• Apr 4th 2010, 11:38 AM
erikzb
limit involving exponential
Hello,

I've lurked this forum before but this is my first time posting. I came across a limit this morning that I feel like I really SHOULD be able to answer but can't. I think I'm missing something stupid. Especially since I think this particular limit comes up a bunch.

$

\lim_{x \to \infty} (1 + \frac{1}{x})^x

$

I've found out that it equals e (from online calculators (Headbang)) but I have no idea why. I've tried converting it into a rational and using l'Hopital's rule, but I'm getting stuck. Am I on the right track? Any hints?
• Apr 4th 2010, 11:41 AM
General
Quote:

Originally Posted by erikzb
I've tried converting it into a rational and using l'Hopital's rule, but I'm getting stuck.

• Apr 4th 2010, 11:47 AM
dwsmith
If we rewrite the limit, this may help.

$[(x+1)/x]^x$

Now lets take the ln and raise to the e

$e^{xln[(x+1)/x]}$

$e^x*[(x+1)/x]$

The limit of $[(x+1)/x]= x/x=1$

Now we have $e^x=e^1=e$
• Apr 4th 2010, 12:11 PM
erikzb
Thanks. I definitely was doing something stupid. I changed the function like in your first step, but then took both the top and bottom the the power x, and changed THOSE to exponentials, and tried using L'H. I'm a crazy person.