Suppose that $f$ is holomorphic on all of $\mathbb{C}$ and that $$\lim_{n\rightarrow \infty} \left(\frac{\partial}{\partial z}\right)^nf(z)$$ exists, uniformly on compact sets, and that this limit is not identically zero. Then the limit function $F$ must be a very particular kind of entire function. Can you say what kind?
My attempt/thoughts:
Well if the limit function is $F$, then $$\frac{\partial}{\partial z}\left(F(z)\right) = \frac{\partial}{\partial z}\left(\lim_{n\rightarrow \infty} \left(\frac{\partial}{\partial z}\right)^nf(z)\right) = \lim_{n\rightarrow \infty} \left(\frac{\partial}{\partial z}\right)^{n+1}f(z) = F(z).$$
Therefore, $$F'(z) = F(z) \implies F(z) = Ae^z$$
Where $A$ is some complex number.
However, this problem is supposed to be one of the harder problems, so my guess is that I must have assumed something that I shouldn't have. Is this wrong? If so, could you point out my error?