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Limit as x approaches 0+ of x^(1/x)

  1. Jan 19, 2013 #1
    1. The problem statement, all variables and given/known data

    limit as x approaches 0+ of x^(1/x)

    2. Relevant equations

    3. The attempt at a solution

    Usually I solve this limit by rewriting the limit as e^log(f(x)) and applying L'hopital rule. However:

    (All limits approach 0+)

    Lim x^(1/x) = exp( Lim log(x^(1/x)) ) = exp( Lim log(x) / x )

    This is where I'd usually apply L'hopital rule and solve the problem. I can't tho, since x tends to 0 and log(x) tends to - infinity. I'm stuck here..

    If anyone could point me in the right direction I'd appreciate.
  2. jcsd
  3. Jan 19, 2013 #2


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    If log(x) approaches -infinity and x approaches 0 and is positive then log(x)/x approaches -infinity. Good idea not to use l'Hopital on it. It's not indeterminant.
  4. Jan 19, 2013 #3
    Oh lol... I did so many exercises with l'Hopital rule that I forgot to check that...
    Thanks a lot!
  5. Jan 19, 2013 #4


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    You did catch it. You realized you can't use l'Hopital. That would have given you the wrong answer! That's a good piece of work right there. If it's not indeterminant the answer should be easy to get. That's all you missed.
    Last edited: Jan 19, 2013
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