Summary: We consider dispersive systems of the form
\[ \begin{aligned}\partial_tU=\Lambda_UU+B_U(U,V),\qquad\varepsilon\partial_tV=\Lambda_VV+B_V(U,U)\end{aligned} \]
in the singular limit \(\varepsilon\rightarrow 0\), where \(\Lambda_U,\Lambda_V\) are linear and \(B_U,B_V\) bilinear mappings. We are interested in deriving error estimates for the approximation obtained through the regular limit system
\[ \begin{aligned}\partial_t\psi_U=\Lambda_U\psi_U-B_U(\psi_U,\Lambda_V^{-1}B_V(\psi_U,\psi_U))\end{aligned} \]
from a more general point of view. Our abstract approximation theorem applies to a number of semilinear systems, such as the Dirac-Klein-Gordon system, the Klein-Gordon-Zakharov system, and a mean field polaron model. It extracts the common features of scattered results in the literature, but also gains an approximation result for the Dirac-Klein-Gordon system which has not been documented in the literature before. We explain that our abstract approximation theorem is sharp in the sense that there exists a quasilinear system of the same structure where the regular limit system makes wrong predictions.