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Singularly perturbed evolution equations may be analyzed by various techniques. In this paper, we focus on ideas of asymptotic analysis and those akin to the Trotter-Kato approximation theorems. Using as an example an abstract telegraph-type system, we show how these two approaches intertwine and complement each other. The paper also may be seen as a continuation of Kato's project (in Turbululence and Navier-Stokes Equations Theory. Proc. Conf. Univ. Paris-Sud, Orsay 1975, Lect. Notes Math., Springer, Berlin, vol 565, pp. 104-112, 1976) taking account of recent results.
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