Today we had an online-test and one of the question was whether the function $z = x+iy \mapsto x$ is differentiable in $0 \in \mathbb{C}$. I thought I'd check it using our definition of complex differentiability and I came to the conclusion, that $\lim_{z \to 0} \frac{\Re(z)}{z}=1.$ I even checked it via wolframalpha afterwards and it told me so as well. However, this was considered to be wrong, because apparently the Cauchy-Riemann equations are not fulfilled, that is to say $1 = \partial_x \Re f(0) \neq \partial_y \Im f(0)=0.$ First, I checked wikipedia to find out what these equations looked like and I found that indeed, "a complex function is differentiable in a point" is equivalent to "a complex function fulfils the Cauchy-Riemann equations". This seems inconsistent to me. I don't see how the limit does not exist. Why does it not exist?
Thanks for any answer in advance.