First, because $e^{ib}$ has modulus (absolute value) $1$ for any real $b$, and $e^x$ is always positive for any real $x$, we have the following identity for any $a\in\Bbb C$:
$\log |e^a|=\log |e^{\mathrm{Re}(a)}e^{i\mathrm{Im}(a)}|= \log e^{\mathrm{Re}(a)}=\mathrm{Re}(a).$
Thus, writing $z=x+iy$, you need to evaluate
$\limsup_{|z|\to\infty} \frac{\mathrm{Re}(-iz)}{|z|}=\limsup_{|z|\to\infty} \frac{y}{\sqrt{x^2+y^2}}=\limsup_{|z|\to\infty} \, (\sin\theta).$
Where $\theta$ is the argument of $z$. Note for any circle $|z|=R>0$ in the complex plane, there is a $z$ for any given angle $\theta\in[0,2\pi)$. Consider the positive imaginary axis...