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Math Help - Complex limit

  1. #1
    Senior Member I-Think's Avatar
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    Complex limit

    Bit confused about a limit
    I need to find this limit

    lim_{z\rightarrow{0}} \frac{Re(z)Im(z)}{|z|^2}
    Wolframalpha gives this limit as 0

    Shouldn't this problem be equivalent

    lim_{(x,y)\rightarrow{(0,0)}} \frac{xy}{x^2+y^2}
    Wolframalpha gives this limit as non-existent
    So now, I'm wondering, what makes these two formulations different?
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  2. #2
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    Re: Complex limit

    \lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^2} is ambiguous because the limit depends on what path you take to approach (0,0). For example, if you approach from the path y = x, you would obtain 1/2, but if you went from the path y = 0, the limit would be zero. Therefore the limit doesn't exist.

    As for the first limit, I haven't studied complex limits but still, you know that z is a complex number but you have no idea which direction z is coming from in order to approach (0,0) or 0 + 0i. However, you know that

    Re(z) = |z|\cos{\theta} and Im(z) = |z| \sin{\theta}

    so \frac{Re(z)Im(z)}{|z|^2} = \frac{|z|^2 \sin{\theta} \cos {\theta}}{|z|^2} = \sin{\theta} \cos {\theta} = \frac{1}{2} \sin {2 \theta}

    Of course, theta is dependent on which path you are taking...if you are traveling on a straight line path towards the origin (e.g. Im(z) = kRe(z)), then theta would be constant and the limit would just be \frac{1}{2} \sin{2 \theta}
    Last edited by richard1234; June 13th 2012 at 10:45 PM.
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  3. #3
    mfb
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    Re: Complex limit

    Should be a bug at WolframAlpha, the limit does not exist.
    I sent a bug report. It can take some months until they fix reported bugs, so I do not expect a response soon.
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  4. #4
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    Re: Complex limit

    WolframAlpha definitely says it's zero (and it even shows steps!):

    Limit[(Re[z]Im[z])&#47 ;(|z|^2), z -> 0] - Wolfram|Alpha

    Hmm...
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  5. #5
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    Re: Complex limit

    Quote Originally Posted by mfb View Post
    Should be a bug at WolframAlpha, the limit does not exist.
    I think "z" is considered as real and in that case the limit is correct. And that is why the limit(x,y) is different.
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