e related limit

**Vamz** · Nov 11th 2010, 04:57 PM

[tex]
\displaystyle \lim_{x \to + \infty} \left( 1 + \frac {3}{x} \right)^{x/2} =
[/tex]

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

$<br /> (1+\frac{1}{x})^{x}<br />$
This needs to be somehow factored out correct?

Can someone please guide me through this?
Much appreciated.

**skeeter** · Nov 11th 2010, 05:02 PM

Originally Posted by Vamz

[tex]
\displaystyle \lim_{x \to + \infty} \left( 1 + \frac {3}{x} \right)^{x/2} =
[/tex]

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

$<br /> (1+\frac{1}{x})^{x}<br />$
This needs to be somehow factored out correct?

Can someone please guide me through this?
Much appreciated.

let ...

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

$x = 2t$

[tex]
\displaystyle \lim_{t \to + \infty} \left( 1 + \frac {3}{2t} \right)^{t} = e^{\frac{3}{2}}
[/tex]

**Vamz** · Nov 11th 2010, 05:20 PM

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!

**Jhevon** · Nov 11th 2010, 05:23 PM

You already asked about this here

skeeter just did a change of variable, and $\displaystyle \lim_{n \to \infty} \left( 1 + \frac xn \right)^n = e^x$

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