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Math Help - limit

  1. #1
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    limit

    find the limit as x approaches 0 of (1/x) to the x power
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  2. #2
    Senior Member yeKciM's Avatar
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    \displaystyle \lim_{x \to 0} (\frac {1}{x})^x = 1
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  3. #3
    Member Jskid's Avatar
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    Judging by the graph of the function the limit does not exist. But it's bean a while since I've calculated a limit and may be wrong

    Actually I'm curious how does \displaystyle \lim_{x \to 0} (\frac {1}{x})^x = 1 ? The function keeps getting more complex each time a derivitive is taken and since log is undefined at 0 you can't use L'Hopital's rule.
    Last edited by Jskid; August 2nd 2010 at 10:33 PM. Reason: fixed spelling of L'Hopital's rule
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  4. #4
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    we need to take the natural log of y=x ln 1/x

    but after that i get confused
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  5. #5
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    then the limit
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by marymm View Post
    find the limit as x approaches 0 of (1/x) to the x power
    \ln \left[ \left(\dfrac{1}{x}\right)^x\right]=-x \ln(x) =- \dfrac{\ln(x)}{1/x}

    Now apply L'Hopitals rule:

    \displaystyle \lim_{x \to 0 } \ln \left[ \left(\dfrac{1}{x}\right)^x\right]=\lim_{x \to 0 } \dfrac{1/x}{1/x^2}=\lim_{x \to 0 } x=0

    So the original limit is 1.

    CB
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