Proving that $f(z) =(2z-z̅ )z̅$ is differentiable at $z = 0$ using the definition of derivative.

Question

Hello I'm currently stuck on solving the derivative of $f(z) = (2z - \bar{z})\bar{z}$. I'm getting stuck at solving $\frac{(2z-\bar{z})\bar{z}}{z}$ as $z \rightarrow 0$

Thanks for any guidance.

Answer 1

Since $\dfrac{\overline z}z$ is bounded and $\lim_{z\to0}2z-\overline z=0$,$$\lim_{z\to0}\frac{\left(2z-\overline z\right)\overline z}{z}=\lim_{z\to0}\left(2z-\overline z\right)\frac{\overline z}z=0.$$