Why will $\lim_{z\to \infty} |e^{z^3}|=0$ only if $\mathrm{Re}(z^3) < 0$?

I have come across a question in a textbook "Sketch the region in an Argand diagram where $\lim_{z\to \infty} |e^{z^3}|=0$

The solution in the book begins "This will only be satisfied if $\mathrm{Re}(z^3)$ is negative", without any justification as if it is obvious, but I cannot see why this is the correct condition.

Can anyone explain please? Thank you!

Solution 1:

If $z=x+iy$ with $x,y\in\mathbb R$, then

$$|\exp(z)| = |\exp(x+iy)|=|\exp(x)\exp(iy)| = \exp(x)\,.$$

Solution 2:

If we denote $z^3=a+ib$, where $a=Re(z^3), b=Im(z^3)$, then $|e^{a+ib}|=|e^ae^{ib}|=|e^a|$. In our case $|e^{a+ib}|=|e^a|\rightarrow 0$ and $e^a$ is a real exponential, so $|e^a|=e^a\rightarrow 0$ for $|a+ib|\rightarrow\infty$ iff $a\rightarrow -\infty$. So we have the necessary and sufficient condition $Re(z^3)\rightarrow-\infty$.