The following theorem is proved: If an analytic function f(z) has singularities only on the real axis and is bounded in magnitude at infinity by a finite but arbitrary power of z, then f(z) has essentially the same limits everywhere at infinity. This theorem enables one to express the contribution from the infinite circle of the Cauchy contour integral in terms of the boundary values of f(z) at infinity along only one of the cuts extending to infinity. The exact dispersion relation is thus determined. As examples, we derive the forward and double pion-nucleon dispersion relations, assuming that the total cross section approaches a finite limit at infinite energy. We see how the subtractions are determined completely by the theorem.