Thread: 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!

No one? I thought these were simple If anyone has any suggestions, please tell me.

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

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

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)