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Homework Help: Use L'Hopital's Rule to relate to limit definition for e

  1. Nov 6, 2012 #1
    1. The problem statement, all variables and given/known data
    It can be shown that
    lim
    n→∞(1 + 1/n)^n = e.
    Use this limit to evaluate the limit below.

    lim
    x→0+ (1 + x)^(1/x)

    2. Relevant equations


    3. The attempt at a solution
    So i guess what i need to do is try to get that limit in the form of the limit definition for e.

    lim
    x→0+ (1 + x)^(1/x)

    x=1/u

    since x-> 0 that means 1/u ->inf

    lim
    x→0+ (1 + 1/u)^(1/(1/u))

    = lim
    1/u→∞ (1 + 1/u)^(u) = e

    I feel like my last 2 steps are wrong, but im sure my answer is right.
     
  2. jcsd
  3. Nov 6, 2012 #2

    SammyS

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    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    You have x = 1/u,

    so if x → 0+, then so does 1/u → 0+.

    What that implies is that u → +∞ .
     
  4. Nov 6, 2012 #3
    That makes more sense. Thank you very much.
     
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