I don't understand why both limits have to be equal in both directions. $\lim_{ \Delta x \rightarrow 0} {\Delta u+i\Delta v \over \Delta x} |_{\Delta y = 0} = \lim_{\Delta y \rightarrow 0}{\Delta u+i\Delta v \over i\Delta y}|_{\Delta x = 0} $
Shouldn't the limit shift in which direction $x ,y$ moves on?