e related limit

• Nov 11th 2010, 04:57 PM
Vamz
e related limit
$$\displaystyle \lim_{x \to + \infty} \left( 1 + \frac {3}{x} \right)^{x/2} =$$

I am simply stumped with this problem.
All I can say, is there is almost an e in there.

$
(1+\frac{1}{x})^{x}
$

This needs to be somehow factored out correct?

Can someone please guide me through this?
Much appreciated.
• Nov 11th 2010, 05:02 PM
skeeter
Quote:

Originally Posted by Vamz
$$\displaystyle \lim_{x \to + \infty} \left( 1 + \frac {3}{x} \right)^{x/2} =$$

I am simply stumped with this problem.
All I can say, is there is almost an e in there.

$
(1+\frac{1}{x})^{x}
$

This needs to be somehow factored out correct?

Can someone please guide me through this?
Much appreciated.

let ...

$\displaystyle t = \frac{x}{2}$

$x = 2t$

$$\displaystyle \lim_{t \to + \infty} \left( 1 + \frac {3}{2t} \right)^{t} = e^{\frac{3}{2}}$$
• Nov 11th 2010, 05:20 PM
Vamz
Thanks. But I still don't really understand how you got there.
I think I understand why you set t=x/2, but after that, you've lost me.

Could you please include the in-between steps?
Thanks!
• Nov 11th 2010, 05:23 PM
Jhevon
skeeter just did a change of variable, and $\displaystyle \lim_{n \to \infty} \left( 1 + \frac xn \right)^n = e^x$