<s>[INST] Evaluate $\lim_{z\to 0} \frac{zRe(z)}{\overline{z}}$ 
$$\lim_{z\to 0} \frac{z\cdot Re(z)}{\overline{z}}$$

$$\lim_{z\to 0} \frac{z\cdot Re(z)}{\overline{z}}=\lim_{(x,y)\to (0,0)} \frac{(x+yi)\cdot x}{x-yi}=\lim_{(x,y)\to (0,0)} \frac{x^2+yxi}{x-yi}\cdot\frac{x+yi}{x+yi}=\lim_{(x,y)\to (0,0)}\frac{x^3+yx^2i+y^2xi-y^2x}{x^2+y^2}=\lim_{(x,y)\to (0,0)}\frac{x^3-y^2x}{x^2+y^2}+i\frac{yx^2+y^2x}{x^2+y^2}$$
How should I continue?
 [/INST] Using your notation:
$$\frac{x(x+iy)}{x-iy}=\frac{x^3-xy^2}{x^2+y^2}+\frac{2x^2y}{x^2+y^2}i$$
and using polar coordinates (why can we? Detail the following):
$$\begin{cases}\cfrac{x^3-xy^2}{x^2+y^2}=r\left(\cos^3\theta-\cos\theta\sin^2\theta\right)\xrightarrow[r\to0]{}0\\{}\\
\cfrac{2x^2y}{x^2+y^2}=r\left(\cos^2\theta\sin\theta\right)\xrightarrow[r\to0]{}0\end{cases}\;\;\implies\frac{ z\,\text{Re}\,z}{\overline z}\xrightarrow[z\to0]{}0$$
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