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Homework Help: Finding a limit help.

  1. Sep 18, 2012 #1
    1. The problem statement, all variables and given/known data

    Finding the limit of (1/n)^(1/ln(n))

    as n-> infinity

    2. Relevant equations

    Log rules

    3. The attempt at a solution

    so I take the ln of both sides and get

    (1/ln(n))*ln(1/n) in an attempt to get it into the proper condition for l'hopitals rule.

    1/ln(infinity) is zero but ln(1/n) is undefined and I have the same problem when trying to multiply by ((1/n)/(1/n)) because it is still an indeterminate form and I can not apply l'hopitals rule.
  2. jcsd
  3. Sep 18, 2012 #2


    Staff: Mentor

    Let y = (1/n)(1/ln(n))
    Then ln y = 1/(ln(n)) * ln (1/n) = 1/(ln(n)) * (-ln(n)) = (-ln(n))/ln(n)

    Now take the limit, noting that for all finite n, the right side above equals -1. Note also that you can switch the order of the lim operation and the ln operation under certain conditions.

    Does that get you started?
  4. Sep 18, 2012 #3
  5. Sep 18, 2012 #4
    Hey thanks ya this does help a lot but I didn't even know about that rule for ln!!
    so ln(1/n) = -ln(n) ?!

    I wonder how many other rules like this there are that I don't even know about haha.

    Thanks so much.
  6. Sep 18, 2012 #5


    Staff: Mentor

    Yes. It's a special case of ln(A/B) = ln(A) - ln(B), with A = 1.
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