[Math] Does $\lim_{z \to 0} \frac {Re(z^2)}{|z|^2}$ exists??

complex numberscomplex-analysislimits

does the limit exist for the following:

$$\lim_{z \to 0} \frac {Re(z^2)}{|z|^2}$$

My take: I tried to substitute $z=x+iy$ for z and then solve the limit but I get 0. According to the wolfram alpha its limit is 1; so can someone please explain this to me? What am I doing wrong here?

Thank You!!!

Best Answer

If $z \in \mathbb R$ and $z \ne 0$, then $\frac {Re(z^2)}{|z|^2}=1$

and

if $z \in i\mathbb R$ and $z \ne 0$, then $\frac {Re(z^2)}{|z|^2}=-1$

Conclusion ?

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