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Math Help - Evaluate the limit

  1. #1
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    Evaluate the limit

    Hi
    Wolfram Alpha has choked on the following:

    \lim_{x\to0}\frac{\ln\cos(x^2)+\sqrt[6]{1+3x^4}-1}{(\sqrt[]{\cos{x}}-\sqrt[4]{e^{-x^2}})\sin{(x^2)}\tan(e^{x^2}-1)}

    I feel powerless over the problem too. Any hints?
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  2. #2
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    That's a lot of input for that online form. I'm having trouble with that also. However, on my copy of Mathematica, it evaluates the limit as \infty in like half a second.

    The way you'd go about proving this by hand would be to use L'Hopital's rule repeatedly, although judging by how contrived the example is, it might require a LOT of work. Maybe there is a smart way, I dunno.

    You can also see that it diverges if you consider the order of every function. Since \cos(x) behaves as 1 near x = 0, we get that \ln(\cos(x)) behaves like \ln(1) = 0 around x =0. Similarly \sin(x^2) behaves like x^2 and \tan(e^{x^2}-1) behaves like e^{x^2}-1. Also, \sqrt{\cos(x)} behaves like \sqrt{1} = 1 around x = 0. If you use these you can see at least justify to yourself that the limit diverges.
    Last edited by Vlasev; August 17th 2010 at 11:04 AM.
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  3. #3
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    Thanks for the answer.
    I think there should be a smart way, since the problem is taken from some math competition. Unfortunately I can't cite the source, because I've found the problem in my handwritten notes.
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