Prove the following limit doesn't exist $\lim_{z \rightarrow 0}(z/\overline{z})^2$
Approach: I am trying to approach different complex numbers and see if I get a different limits. I am also trying to approach this in polar coordinates, but I think it's useless because as a complex number approaches 0, the angle from the positive axis shouldn't change. All the complex number I have tried yield to the same result.
Best Answer
Hint:
1) When $z = a$ , $(z/\overline{z})^2 = (z/z)^2 = 1^2 = 1$.
2) When $z = a+ai$, $(z/\overline{z})^2 = (\frac{a(1+i)}{a(1-i)})^2 = (\frac{(1+i)^2}{2})^2 = (\frac{2i}{2})^2 = i^2 = -1$.
Here $a \in \mathbb R$
Generalisation:
Let $z = Re^{\theta i}$, where $R \in \mathbb R$, $\overline{z} = Re^{-\theta i}$, $\lim_{z \rightarrow 0}(z/\overline{z})^2= \lim_{R \rightarrow 0}(\frac{Re^{\theta i}}{Re^{-\theta i}})^2 = \lim_{R \rightarrow 0}(e^{2\theta i})^2 = \lim_{R \rightarrow 0}(e^{4\theta i}) = \cos 4 \theta + i\sin 4 \theta $.
Substituting $\theta$ proper values, and one can get infinitly counterexample.