# Thread: limit involving exponential

1. ## 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.

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

I've found out that it equals e (from online calculators ) 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?

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

3. If we rewrite the limit, this may help.

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

Now lets take the ln and raise to the e

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

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

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

Now we have $\displaystyle e^x=e^1=e$

4. 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.

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