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Math Help - complex limit

  1. #1
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    complex limit

    Suppose  f(z) = e^{1/z} . What is  \lim\limits_{x=0, y \to 0} |f(z)| ?

    So  \lim\limits_{x=0, y \to 0} |f(z)| = \lim\limits_{ y \to 0} |e^{1/iy}| =  \lim\limits_{x=0, y \to 0} |\cos (1/y)-i \sin(1/y)| which does not exist.

    Is this correct?
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  2. #2
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    Quote Originally Posted by manjohn12 View Post
    Suppose  f(z) = e^{1/z} . What is  \lim\limits_{x=0, y \to 0} |f(z)| ?

    So  \lim\limits_{x=0, y \to 0} |f(z)| = \lim\limits_{ y \to 0} |e^{1/iy}| =  \lim\limits_{x=0, y \to 0} |\cos (1/y)-i \sin(1/y)| which does not exist.

    Is this correct?
    No because |\cos (1/y)-i \sin(1/y)| = \sqrt{cos^2(1/y)+sin^2(1/y)} = 1

    If you prefer \lim\limits_{ y \to 0} |e^{1/iy}| = \lim\limits_{ y \to 0} |e^{-i/y}| =  \lim\limits_{ y \to 0} 1 = 1
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