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Let us teach this generalization of the final-value theorem

A suggestion relevant to teaching the use of Laplace transforms in a basic course of engineering mathematics (or circuit theory, automatic control, etc) is made. The useful ‘final-value’ theorem for a function f(t), , , makes sense only if , , exists. A generalization of this theorem for time functions for which does not exist, but the time average exists, states that as , . This generalization includes the case of periodic or asymptotically periodic functions, and almost-periodic functions that can be given by finite sums of periodic functions. The proofs include the case of f(t) tending to fas(t) exponentially, which is realistic for the main physics and circuit applications. Extension of the results to discrete sequences, treatable by the z-transform, is briefly considered. The generalized form of the final-value theorem should be included in courses of engineering mathematics. The teacher can introduce interesting new problems into the lesson, and provide a better connection with the (usually later) study of the Fourier series and Fourier transform.