The Final Value Theorem Revisited - Infinite Limits and Irrational Functions

The aim of this article is to publicize and prove the ";infinite-limit"; version of the final value theorem. The version we provide is a slight refinement of the classical literature in that we require that s approach zero through the right-half plane to obtain the correct sign of the infinite limit. We first consider the case of rational Laplace transforms and then state a version that applies to irrational functions. For rational Laplace transforms with poles in the OLHP or at the origin, the extended final value theorem provides the correct infinite limit. For irrational Laplace transforms, the generalized final value theorem provides the analogous result. Finally, we point to a detailed analysis of the final value theorem for piecewise continuous functions.