# Thread: Limits on Complex Numbers

1. ## Limits on Complex Numbers

Here are two from my homework I can't quite figure out.

$1. \lim_{\Delta z \to 0} \frac {(z_0 + \Delta z )^3 - z_0^3}{\Delta z}$

$2 \lim_{z \to 2\pi} (e^{iz} - e^{-iz})$

2. Originally Posted by Creebe
Here are two from my homework I can't quite figure out.

$1. \lim_{\Delta z \to 0} \frac {(z_0 + \Delta z )^3 - z_0^3}{\Delta z}$

$\lim_{\Delta_z \to 0} \frac {(z_0 + \Delta_z )^3 - z_0^3}{\Delta_z}=:f'(z_0)\,,\,with\,\,\,f(z)=z^3$, or directly:

$\lim_{\Delta_z \to 0} \frac {(z_0 + \Delta_z )^3 - z_0^3}{\Delta_z}=\lim_{\Delta_z\to 0}\frac{3z_0^2\Delta_z+3z_0\Delta^2_z+\Delta^3_z}{ \Delta_z}=\lim_{\Delta_z\to 0}\left(3z_0^2+3z_0\Delta_z+\Delta^2_z\right)=3z_0 ^2$

$2 \lim_{z \to 2\pi} (e^{iz} - e^{-iz})$

Hint: $e^{iz}-e^{-iz}=2i\sin z\,,\,\,z\in\mathbb{C}$

Tonio