Here is the question:
Let $f$ be a meromorhic function on $\mathbb{C}$, having poles at the following three points: $z=5$, $z=1+3i$ and $z=3-4i$. Also, let $f$ have one removable singularity at $z=3$. For the following, find the value or explain why not enough information is given to find the quantity.
a) $\lim_{z\rightarrow5}|f(z)|$
b) $\lim_{z\rightarrow1+2i}(z-1-2i)f(z)$
c) $\lim_{z\rightarrow\infty}|f(z)|$
My thoughts:
I was wondering if I would be able to write $f$ as a rational function, such as $f(z)=\frac{(z-3)^m}{(z-3)^m(z-5)^n(z-(1+3i))^k(z-(3-4i))^l}$ for $m, n, k, l\in\mathbb{Z^{+}}$? This just doesn't feel right, because I am not really sure how this would help me do $(a)$ or $(b)$.
For $(c)$, wouldn't the limit just be $0$ (I suppose if I can write $f$ in the above form).
My other idea was to try and write $f$ as a Laurent series, but I am not quite sure how to "give" $f$ the removable singularity as well as all the poles. I suppose I could try and think of a Laurent series that would satisfy the conditions, prove that it satisfies the conditions, and then try and find $(a), (b), (c)$, but I am not sure if this would be the most efficient way, or if there would be another way. I appreciate any ideas, thoughts, etc. Thank you!
Consider the fact that if $f:\mathbb C\backslash S\to\mathbb C$ is a function satisfying your given conditions, where $S$ is the set containing the four given singularities, and $g$ an entire function, then $\tilde f:=f+g$ also satisfies the given conditions, while possibly resulting in different limits. I recommend not to try to find a general term describing your function, and instead relying on general facts about meromorphic functions and their singularities.
a) There's a theorem which says that an isolated singularity $z_0$ is a pole iff $\lim_{z\to z_0}\vert f(z)\vert=\infty$.
b) I think you're missing a $z$. And maybe you wanted to make it 3, not 2? Is it supposed to be $\lim_{z\to(1+3\mathrm i)}(z-1-3\mathrm i)f(z)$? If yes, remember the definition of the order of a pole, and consider that you don't know the order.
c) You found an example where this limit is $0$ (though you should really remove the numerator in your fraction for it to work). Now add an arbitrary entire function as I did above. Is the limit still $0$?