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Math Help - multivarible limits using polar coordinates

  1. #1
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    multivarible limits using polar coordinates

    I'm stuck on:

    \lim_{(x,y) \rightarrow (0,0)} (x^2+y^2)ln(x^2+y^2)

    so far I have:

     ((rcos \theta)^2+(rsin \theta)^2)(ln((rcos \theta)^2+(rsin \theta)^2)

    = (r^2)(ln(r^2))

    at which point I'm stuck.
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  2. #2
    Jen
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    Quote Originally Posted by lllll View Post
    I'm stuck on:

    \lim_{(x,y) \rightarrow (0,0)} (x^2+y^2)ln(x^2+y^2)

    so far I have:

     ((rcos \theta)^2+(rsin \theta)^2)(ln((rcos \theta)^2+(rsin \theta)^2)

    = (r^2)(ln(r^2))

    at which point I'm stuck.
    Use L'Hospital's rule,

    Re-write as

    (r^2)(ln(r^2))=\frac{2ln(r)}{\frac{1}{r^2}}

    By L.H. rule we get,

    \frac{\frac{2}{r}}{\frac{-2}{r^3}}=-r^2

    lim as r goes to zero is zero.

    Hope that helps.
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