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Math Help - Limit of zero to the zero

  1. #1
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    Limit of zero to the zero

    \lim\limits_{n \to \infty} \left(\frac{1}{n}\right)^\frac{1}{n}. The answer is 1, but how do I show this?

    My textbook says for this scenario take y = f(x)^{g(x)}, take \ln y, find the limit of that, and then the limit of y = e^{\ln y}. I'm not sure how to do this one either: \lim\limits_{n \to \infty} \ln \left[\left(\frac{1}{n}\right)^\frac{1}{n}\right]

    EDIT: OK solved.

    \lim\limits_{n \to \infty} \ln \left[\left(\frac{1}{n}\right)^\frac{1}{n}\right] = \lim\limits_{n \to \infty} \frac{\ln \frac{1}{n}}{n} which is solved with l'hopital's rule, comes to zero, and the total limit comes to one. thanks!
    Last edited by VinceW; August 21st 2012 at 06:06 AM.
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  2. #2
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    Re: Limit of zero to the zero

    Quote Originally Posted by VinceW View Post
    \lim\limits_{n \to \infty} \left(\frac{1}{n}\right)^\frac{1}{n}. The answer is 1, but how do I show this?

    My textbook says for this scenario take y = f(x)^{g(x)}, take \ln y, find the limit of that, and then the limit of y = e^{\ln y}. I'm not sure how to do this one either: \lim\limits_{n \to \infty} \ln \left[\left(\frac{1}{n}\right)^\frac{1}{n}\right]

    EDIT: OK solved.

    \lim\limits_{n \to \infty} \ln \left[\left(\frac{1}{n}\right)^\frac{1}{n}\right] = \lim\limits_{n \to \infty} \frac{\ln \frac{1}{n}}{n} which is solved with l'hopital's rule, comes to zero, and the total limit comes to one. thanks!
    There is as much material to suggest that 0^0 = 0. You have done the limit correctly (as far as I can see) but the problem 0^0 has been labelled as undefined.

    -Dan
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  3. #3
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    Re: Limit of zero to the zero

    Hi VinceW !

    Just have a look at this paper and you will see several commented examples of 0^0 :
    "Zéro puissance zéro - Zero to the zero power"
    Scribd
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