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Math Help - Need help with two (probably simple) limits

  1. #1
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    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!
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  2. #2
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    No one? I thought these were simple If anyone has any suggestions, please tell me.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by phack View Post
    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
    Attached Thumbnails Attached Thumbnails Need help with two (probably simple) limits-lim1a.gif  
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by phack View Post
    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
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by phack View Post
    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)
    Last edited by CaptainBlack; April 10th 2007 at 11:29 PM.
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