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Math Help - Limit problems

  1. #1
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    Limit problems

    Hello ,

    I was solving problems on limits when i encountered this one with a complex number.

    limit Z->0 (Z*/Z) WHERE Z*is conjugate of Z.

    If i get 0/0 i used to factorize numerator, which almost always cancels with the denominator ,to get the answer which i could not do for this one.The problem for me is Z has two variables x and y and i dont know how to apply limits for them.I think the answer should be 'undefined' but i cant prove it mathematically.Any simple way to solve these kinds of problems?
    Moreover are there any general ways of solving these limit problems?

    Thank you
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  2. #2
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    Quote Originally Posted by essex View Post
    Hello ,

    I was solving problems on limits when i encountered this one with a complex number.

    limit Z->0 (Z*/Z) WHERE Z*is conjugate of Z.

    If i get 0/0 i used to factorize numerator, which almost always cancels with the denominator ,to get the answer which i could not do for this one.The problem for me is Z has two variables x and y and i dont know how to apply limits for them.I think the answer should be 'undefined' but i cant prove it mathematically.Any simple way to solve these kinds of problems?
    Moreover are there any general ways of solving these limit problems?

    Thank you
    Let z = x + iy.

    Then you want \lim_{(x, y) \rightarrow (0, 0)} \frac{x - iy}{x + iy}.

    Since the value depends on how the limit is taken, the limit does not exist.
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  3. #3
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    Thank you mr fantastic .So we have to find the value as x approaches 0 (we get -1) and find the value as y approaches 0 (we get 1).Since both are different it doenot exist.

    Is this the way we should do it for a number with different variables?
    (i.e) by letting each variable of the number go to the given limit seperately and check if they are equal ?
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  4. #4
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    Quote Originally Posted by essex View Post
    Thank you mr fantastic .So we have to find the value as x approaches 0 (we get -1) and find the value as y approaches 0 (we get 1).Since both are different it doenot exist.

    Is this the way we should do it for a number with different variables?
    (i.e) by letting each variable of the number go to the given limit seperately and check if they are equal ?
    You should consider different paths and see what happens. eg If (x, y) \rightarrow (0, 0) along the path y = mx:

    \lim_{x \rightarrow 0} \frac{x - imx}{x + imx} = \lim_{x \rightarrow 0} \frac{1 - im}{1 + im} = \frac{1 - im}{1 + im} which clearly depends on m.
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  5. #5
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    Oh ok. So it depends on the path given ,but if theres no specific path then we find it seperately for different variables and check if they are equal.correct?

    Thank you very much
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