[Math] Show the limit $\lim_{z\to0} \frac{2|z|^2}{\operatorname{Re}(z^2)}$ does not exist

Question

Complex variables:
How do I prove that $$\lim_{z\to0} \frac{2|z|^2}{\operatorname{Re}(z^2)}$$ does not exist?

I am sorry, I just started learning limits in complex variables, And I am very confused about limits
Any help would be appreciated
Thank you

Answer 1

Take $z=re^{i \theta}$ (where $r=|z|$)

When $r \longrightarrow 0 $ then $z \longrightarrow0$

If your limit, after this substitution, depends on the values of $\theta$ then does not exist.