# limits to infinity

• Nov 29th 2009, 02:58 AM
alexandrabel90
limits to infinity
how do you find
lim of (1 + 2/x)^x as x tends to infinity?
• Nov 29th 2009, 03:16 AM
tonio
Quote:

Originally Posted by alexandrabel90
how do you find
lim of (1 + 2/x)^x as x tends to infinity?

If you already know that $\lim_{x\to\infty}\left(1+\frac{1}{f(x)}\right)^{f( x)}=e\,,\mbox { for any function }f(x)\mbox{ s.t. } f(x)\xrightarrow [x\to\infty] {}\infty \mbox{ whenever } x\rightarrow \infty$ ,then:

$\lim_{x\to\infty}\left(1+\frac{2}{x}\right)^x=\lim _{x\to\infty}\left[\left(1+\frac{1}{x\slash 2}\right)^{x\slash2}\right]^2=e^2$

Tonio