the singularity at infinity
On the Riemann sphere, infinity is just another point — the north pole, the place all the lines of the plane rush toward. So it is natural to ask: how does a function behave out there, at infinity? Does it settle down, blow up, or go wild? Classifying a function's behaviour at infinity is exactly the same problem as before, viewed through a simple change of variable.
The trick is to substitute w = 1/z, which turns the point at infinity (z very large) into the point w = 0 (w very small). Define g(w) = f(1/w) and study g at 0. Whatever kind of isolated singularity g has at w = 0, that is by definition the kind of singularity f has at infinity: removable means f tends to a finite limit at infinity (g is bounded near 0); a pole of order m at w = 0 means f grows like z^m at infinity; an essential singularity at w = 0 means f behaves wildly at infinity. You read off the type from the Laurent series of f in powers of 1/z for large |z|, just as you would near any finite point.
This viewpoint completes the picture and is genuinely useful. A polynomial of degree n has a pole of order n at infinity (it grows like z^n), which is why polynomials are not bounded and Liouville does not contradict them. e^z has an essential singularity at infinity (set w = 1/z and look at e^(1/w) at 0). A function holomorphic on the whole sphere, infinity included, must be constant; and a function meromorphic on the whole sphere is exactly a rational function. Counting the singularity at infinity is also what makes 'the sum of all residues, including the one at infinity, is zero' come out cleanly.
Classify f(z) = z^2 + 1/z at infinity. Set w = 1/z: g(w) = f(1/w) = 1/w^2 + w. Near w = 0 this has principal part 1/w^2 — a pole of order 2. So f has a pole of order 2 at infinity (it grows like z^2). For comparison, f(z) = sin z gives g(w) = sin(1/w), essential at 0, so sin z has an essential singularity at infinity.
Behaviour at infinity = behaviour of f(1/w) at w = 0; here z^2 + 1/z has a double pole at infinity.
Watch the bookkeeping: f bounded at infinity is a removable singularity there, but 'removable at infinity' still allows a nonzero finite limit; and a polynomial of positive degree has a POLE at infinity, not an essential one, despite being unbounded.