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Finding the limit of a function with a complex exponent

  1. Oct 14, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]\lim_{z \to 0} (\frac{sinz}{z})^{1/z^2}[/itex]

    where z is complex

    2. Relevant equations

    The standard definition of a limit
    L'hopital's rule?

    3. The attempt at a solution
    I'm quite stumped by this one. There doesn't seem to be a way to break it down into different limits or even to manipulate it much algebraically. Wolfram says the answer is [itex]e^{-1/6}[/itex] but I'm not sure how to arrive at it. I tried starting by proving that that is the limit if z is real and then extending it to the complex plane, but I can't even solve it for that case. I feel like there's an obvious theorem or something along those lines that I am forgetting. Any hints?
  2. jcsd
  3. Oct 14, 2011 #2


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    Homework Helper

    1) log it
    2) find the limit. It helps to know what (sinz)/z approaches as z approaches 0. Can we apply l'hopital's rule?
    3) reverse the log.
  4. Oct 14, 2011 #3
    It worked! Thanks, that was a really clever and elegant solution
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