I have the task of adapting L'Hospital's rule to complex-functions $f(x)$ and $g(x)$, where $x \in \mathbb{R}$. The new theorem should look like:
Suppose $f$ and $g$ are complex-valued functions and are differentiable in $(a,b)$, and $g'(x) \neq 0$ for $x \in (a,b)$, where $-\infty \leq a < b \leq \infty$. Suppose$$ \lim_{x \to a}\frac{f'(x)}{g'(x)} = A. \\ \lim_{x \to a}\frac{\mathfrak{R}g'(x)}{\mathfrak{I}g'(x)} \text{ or } \lim_{x \to a}\frac{\mathfrak{I}g'(x)}{\mathfrak{R}g'(x)} \text{ exists}$$ If $$ \lim_{x \to a}f(x) = 0 = \lim_{x \to a}g(x)\\ \text{or} \\ \lim_{x \to a}g(x) = \infty$$ then $$\lim_{x \to a}\frac{f(x)}{g(x)} \to A$$
Given that we know that it holds for vector-valued functions in the case where $\mathbf{f}(x)$ becomes a vector-valued function whereas $g(x)$ remains a real-valued function (it was the previous part of the problem), how do we apply that here? We can multiply the numerator and denumerator with the conjugate to get a vector-valued function and real-valued function as numerator and denumerator, respectively. But we end up with the ugly $$\frac{f'\bar{g}'}{\left \vert g'(x) \right \vert^2} = \frac{f'(x)}{g'(x)}.$$I've tried endlesss algebraic manipulation of the LHS and I've come up with nothing useful. I've already looked online, which has led to papers written by Gary Lawlor and D.S. Carter. But both use tools out of the scope of the problem's section. And I also know there are many L'Hospital's for Complex Function questions on here, but all of them use tools in complex analysis and this is just for Baby Rudin.