Residue at infinity for different functions

Question

My main question is how can I identify that f(z)=(z+i)^-1 or other functions have a non-zero residue at infinity. The reason for confusion is I am taught that if f(z)~z^-2 as z tends to infinity then we have a removable singularity at z=infinity. Would this tells us that if f(z)~z^-1 as z tends to infinity, then this function would have a simple pole a infinity for all function that tend the same way and what would this mean about functions that tend off as z^-3 etc. Hope this question makes sense.