Limit of complex number over its conjugate as it approaches zero Basically, I have to look at this thing:
$$\lim_{z\to 0}{\frac{z}{z^*}}=1$$
I know this, but I need to prove it. I tried using the following approach.
$$\frac{re^{i\theta}}{re^{-i\theta}}=e^{2i\theta}$$
Unless I can get a definite value for $\arg 0$, I'm stuck at an impasse, as this makes it clearly look like the limit doesn't even exist since the angle of attack for this limit changes its value. What am I missing? I have to be missing something here in this hot mess.
 A: You are correct. The fraction is dependent to $\theta$ and the limit does not exist and is not equal to $1$.
A: The way you are proceeding is completely correct. Just add the conclusion.
We have $$\lim_{z\to 0}{\frac{z}{z^*}}$$
Now Z is in the complex plane. Let's move toward zero through the imaginary axis.
as on imaginary axis, z is imaginary always, so $z=-z^*$
Hence  $$\lim_{z\to 0}{\frac{z}{z^*}}=\frac{z}{-z}=-1$$
Now let's approach zero from the real axis, on the real axis we have
$z=z^*$
Hence  $$\lim_{z\to 0}{\frac{z}{z^*}}=\frac{z}{z}=1$$
Conclusion: If we approach zero along different paths then we get different limit hence the limit at zero doesn't exist.

In your case: The way you proceeded, 
We know that that $z\to 0$ is to say $r\to 0$, and no condition on $\theta$
$$\begin{align}
\frac{z}{z^*} = e^{-2i\theta}
\end{align}$$
But here limit depends on $\theta$, that is depends on how z approaches zero. In another word, if we choose the different direction of approach we'll get different values of limit.
Note: the different value of limits that we are getting will always lie on a circle of unit radius around the center.
