Definition: 1stly, each of the expressions $e^{\frac i z}$, $\frac{Re(z)}{|z|^2}$ and $\frac{Im(z)}{|z|^2}$ are given to be defined on the respective maximal possible subsets of $\mathbb C$. I think each maximal subset is $\mathbb C \ \setminus 0$.
Conjecture: I suspect $\lim_{z \to 0} e^{\frac i z}$ doesn't exist using the non-existence of either of the ff limits $\lim_{z \to 0} \frac{Re(z)}{|z|^2}$ (or $\lim_{z \to 0} \frac{Im(z)}{|z|^2}$), but I think it's not necessary to use both of the latter.
Proof:
I think $\lim_{z \to 0} \frac{Re(z)}{|z|^2}$ doesn't exist because of the same reason as in the $\mathbb R^2$ case:
Along $x=0$, $\frac{Re(z)}{|z|^2}$ is identically zero.
Along $y=0$, $\frac{Re(z)}{|z|^2} = \frac 1 x$, w/c does not approach anything as $x \to 0$ (and we have $x \to 0$ from $z \to 0$).
Similar for $\lim_{z \to 0} \frac{Im(z)}{|z|^2}$.
As for $\lim_{z \to 0} e^{\frac i z}$, we have $$e^{\frac i z} = e^{\frac{Im(z)}{|z|^2}}[\cos(\frac{Re(z)}{|z|^2}) + i \sin(\frac{Re(z)}{|z|^2})]$$ $$= e^{\frac{y}{x^2+y^2}}[\cos(\frac{x}{x^2+y^2}) + i \sin(\frac{x}{x^2+y^2})]$$
$$= e^{\frac{y}{x^2+y^2}}\cos(\frac{x}{x^2+y^2}) + i e^{\frac{y}{x^2+y^2}}\sin(\frac{x}{x^2+y^2})$$
Therefore, $\lim_{z \to 0} e^{\frac i z}$ almost definitely doesn't exist because each of its real and imaginary parts are composed of $e^{\text{does not exist}}$ ($\cos$ or $\sin$) ($\text{does not exist}$).
QED
Questions:
Did I make any mistakes?
If not, then what's going on here?
Why would we ever think $\lim_{z \to 0} e^{\frac i z}$ exists? There's too much evidence to the contrary.
Guess for #3: It is possible somehow that, say for the real part, both of these limits don't exist
$$\lim_{z \to 0} e^{\frac{y}{x^2+y^2}}, \lim_{z \to 0}\cos(\frac{x}{x^2+y^2}),$$
but then somehow
$$\lim_{z \to 0}e^{\frac{y}{x^2+y^2}}\cos(\frac{x}{x^2+y^2})$$ might exist?
