# Need help with two (probably simple) limits

• Apr 10th 2007, 01:52 PM
phack
Need help with two (probably simple) limits
I'm having some difficulty with these two limits. I learned this stuff a while back, but as you know, you sometimes forget the simple stuff after you havent done it in a while. :(

1:

limit as x goes to infinity of [(x+1)(x+1)^x]/(x^x)

*note, only the second x+1 is raised to the power of x in the numerator

2:

limit as x goes to infinity of [x/(x+1)]^x

Thanks!
• Apr 10th 2007, 03:56 PM
phack
No one? I thought these were simple:( If anyone has any suggestions, please tell me.
• Apr 10th 2007, 04:10 PM
Jhevon
Quote:

Originally Posted by phack
I'm having some difficulty with these two limits. I learned this stuff a while back, but as you know, you sometimes forget the simple stuff after you havent done it in a while. :(

1:

limit as x goes to infinity of [(x+1)(x+1)^x]/(x^x)

*note, only the second x+1 is raised to the power of x in the numerator

Here
• Apr 10th 2007, 08:49 PM
CaptainBlack
Quote:

Originally Posted by phack
I'm having some difficulty with these two limits. I learned this stuff a while back, but as you know, you sometimes forget the simple stuff after you havent done it in a while. :(

1:

limit as x goes to infinity of [(x+1)(x+1)^x]/(x^x)

*note, only the second x+1 is raised to the power of x in the numerator

Look at what happens to the log of f(x)=[(x+1)(x+1)^x]/(x^x).

log(f(x)) = log(x+1) + x log(x+1) -x log(x)

............= log(x+1) + x log(1+1/x))

The first term goes to infty as x goes to infty, and whatever the second term
does it is positive for positive x and so log(f(x)) goes to infty as x goes to
infty and hence so does f(x).

RonL
• Apr 10th 2007, 08:55 PM
CaptainBlack
Quote:

Originally Posted by phack
2:

limit as x goes to infinity of [x/(x+1)]^x

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

Now you should know that lim_{x->infty} [(1+1/x)^x] = e, so:

lim_{x->infty} [x/(x+1)]^x = 1/e.

RonL

(note: While the limit in Jhevon's post is correct, this shows that there is an error
in the derivation)