1. ## 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?

2. Originally Posted by manjohn12
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 $\lim\limits_{ y \to 0} |e^{1/iy}| = \lim\limits_{ y \to 0} |e^{-i/y}| = \lim\limits_{ y \to 0} 1 = 1$