Residues

(RefLink[Limit, paclet:ref/Limit])[expr,x->(Subscript[x, 0])] tells you what the value of expr is when x tends to (Subscript[x, 0]). When this value is infinite, it is often useful instead to know the residue of expr when x equals (Subscript[x, 0]). The residue is given by the coefficient of (((x-(Subscript[x, 0])))^(-1)) in the power series expansion of expr about the point (Subscript[x, 0]).

Computing residues.

The residue here is equal to 1:

Residue[1/x,{x,0}]

The residue here is zero:

Residue[1/x^2,{x,0}]

Residues can be computed at the point at infinity:

Residue[1/x,{x,ComplexInfinity}]