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Math Help - Comparing growth of functions

  1. #1
    Junior Member
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    Comparing growth of functions

    I have two functions:

    f(n)=3^\log_{2}n^2
    and
    g(n)=3^\log_{2}\frac{n^2}{2}

    I have to prove it its big-Oh/theta/omega

    I have to use the limit method where:
    limit n going to infinity of
    \frac{f(n)}{g(n)}
    Then L'Hospital's rule, but when I take the derivatives of both and cancel out everything I just get what I was left with before. Am I doing something wrong? Am I not taking the derivative correctly?
    I was using \frac{d}{dx}a^x=log(a)a^x and chain rule.
    Can someone help me to clarify this?
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  2. #2
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    That L'Hospital's rule leads to the same ratio is natural since, as you pointed out, the derivative of a^x again contains a^x.

    One can easily do without L'Hospital's rule here since f(n)/g(n) is a constant.
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  3. #3
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    Oh, it isn't infinity over infinity?
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  4. #4
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    It is. However, for instance, 3n^2/(2n^2) is also infinity over infinity, and yet this fraction not only tends to 3/2, but is equal to 3/2.

    I personally was taught to avoid L'Hopital's rule and to study the function's behavior. Often this results in better understanding of why the limit is what it is.
    Last edited by emakarov; October 14th 2010 at 03:25 AM. Reason: Typo
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