<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] Bounding the numerators of real and imaginary parts:
$$|x^3-y^2x|\le|x|(x^2 + y^2),$$
$$|yx^2+y^2x|\le\max\{|x|,|y|\}(x^2 + y^2).$$
So the limit is zero.
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