This is from my notes on the derivation of the Cauchy Riemann equations. My question is about going from the limit as z->z_0 to the lim as Δx->0. Why is f'(z_0) the same as the partial derivative of f with respect to x? If we say f(z)=u+iv where u and v are functions of x and y, then wouldn't f'(z_0) be different from the partial derivative with respect to x?