# Extract real part of a complex function

I have a problem for extracting the real part of a complex number. The problem is the following: Suppose $$f(z)=\frac{1}{z+\frac{\Delta_{1}}{z+\frac{\Delta_{2}}{z+\frac{\Delta_{3}}{z+...}}}}$$, in which z is a complex number $$(z= a+bi)$$ and $$\Delta_{1}$$, $$\Delta_{2}$$ ... are number and $$f(z)$$ is a continued fraction. And I want to compute $$\phi(b)$$ since I would like to plot $$\phi(b)$$ in the end

$$\phi(b)=\lim_{a\rightarrow0}Ref(a+bi)$$

But Mathematica doesn't allow me to do that. I use "Limit" in Mathematica and set all $$\Delta_{n} = 0 , n > 1$$ and $$\Delta_{1}=1$$ , I get 0. Even I let some $$\Delta_{n}$$ to be nonzero, I still end up with 0.

Here is my code. I set $$\Delta_{1}=1$$ and $$\Delta_{2}=0.5$$, rest are zero.

Limit[ComplexExpand[Re[1/(a + b*I + (1/(a + b*I + (0.5/(a + b*I)))))]], a -> 0]


Can someone help me with this ? I am guessing the "Limit" function in Mathematica can't not handle this problem.

Thank you very much !!!

Are you sure 0 is not the right answer? The problem is not with the limit. Look at your function without limits. I changed your 0.5 to 1/2.
ComplexExpand[Re[1/(a + b*I + (1/(a + b*I + (1/2/(a + b*I)))))]] // Simplify

with a as a factor in the numerator you don't need the limit.
% /. a -> 0