technique for computing residues

The following two facts are quite useful for computing residues:

If $f$ has a pole of order at most $n+1$ at $x$, then

$$\mathrm{Res}(f;x)=\underset{y\to x}{lim}\frac{1}{n!}\frac{d^n}{d{y}^n}\left({(y-x)}^{n+1}f(y)\right).$$

If $g$ is regular at $x$ and $f$ has a simple pole at $x$, then $\mathrm{Res}(fg;x)=g(x)\mathrm{Res}(f;x)$.