I have to write in theory the derivation of a complex variable.

So, when

$\displaystyle \Delta x=0$

$\displaystyle \Delta z=i\Delta y$

i have

$\displaystyle \lim_{\Delta z \rightarrow 0}\frac{f(z+\Delta z)-f(z)}{\Delta z}=\lim_{\Delta y \rightarrow 0} \frac{-i\Delta y}{i\Delta y}=-1$

what do we do to get$\displaystyle \lim_{\Delta y \rightarrow 0} \frac{-i\Delta y}{i\Delta y}$ from $\displaystyle \lim_{\Delta z \rightarrow 0}\frac{f(z+\Delta z)-f(z)}{\Delta z}$