this works. You can save a few steps by noting that
$|\bar{z}|=|z|$
You might also consider using polar form.
If $z=r\exp(i\theta)$ then $\overline{~z~}^3=r^3\exp(-3i\theta)$
Form which you get $\dfrac{\overline{~z~}^3}{z}=r^2\exp(-4i\theta)$ where $\theta\to 0~\&~r\to 0$