In contrast to the use of white noise in stochastic models, wideband noise is more realistic for various applications. White noise is a good process to work with, but it is difficult to realize in real applications, whereas wideband noise is a good approximation to white noise. This work is focused on functional differential equations subject to wideband noise. By virtue of the ideas of functional derivatives developed in recent work on functional It\^o formula, using perturbed test function methods combined with martingale techniques, this paper shows that when the small parameter tends to zero, the underlying process converges to a limit that is governed by solutions of a stochastic functional differential equation. This paper also gives an integro-differential system with wideband noise as an example and examines its asymptotic properties. Not only are the results interesting from a mathematical point of view, but also they will be of great utility to a wide range of problems in control and optimization problems.