I'm trying to show that
$$\lim_{n \to \infty}\int_{\gamma_n}\frac{\pi \cot \pi z}{(u+z)^2}dz = 0$$
goes to zero, where $\gamma_n$ is a circle of radius $R = n+\frac{1}{2}$ and $n\in\mathbb{Z}$. $u\in\mathbb{C\setminus\mathbb{Z}}$ was given in an earlier problem to compute the integral around $\gamma_n$ for a fixed $n$ using the residue theorem and is rather uninteresting for my current problem. Given that this circle only intersects with the real numbers at $\pm(n+\frac{1}{2})$, $f(z)=0$ in these points (because of the cotangent).
I am not at all certain, however, how cotangent behaves on the rest of the complex plane, and therefore, how I could calculate this limit. If cotangent were bounded somehow (i.e. sin(z) had a lower bound), I could argue that taking n (that is, the radius of the circle) to infinity would cause the whole limit to go to zero. I'm not sure about this, and I'm inclined to say there is no such lower bound for sin(z). I could be wrong though. Maybe someone could confirm this? Also, suggestions on how to compute this limit would be greatly appreciated.