Residue theorem and laurent expansion

[elimenohpee]

Homework Statement

I need to calculate the residue of a function at infinity. My teacher does this by expanding the function in a laurent expansion and deduces the value from that. That seems much harder than it needs to be. For example, in the notes he calculates the residue at infinity of:
$$g(z)=\frac{\sqrt{z^{2}-1}}{z-t}=...=-t$$

Is there an easier way than resorting to a laurent series? If I let z approach infinity in the function above, I get 1 not -t, so I'm assuming you can't evaluate the residue in that way?

Specifically I need to find the residue at infinity of
$$f(z)=\frac{z^{2}}{\sqrt{z^{2}-1}(z-t)}$$
but I'm looking for a method to do so.

 

[Klockan3]

If you have nicer functions than that there are a few tricks, but when the Laurent expansion doesn't terminate I don't think that there is an easier way than to use a series expansion.