We consider linear, time-invariant state-space systems under high-gain state feedback. The analysis is couched in terms of singular system theory and Grassman manifolds. Our work is distinguished from that of other authors by the fact that we do not allow a gain-dependent state coordinate change. Simple necessary and sufficient conditions are proven under which a singular system is a high-gain limit of a given state-space system. It is shown that the feedback matrix achieves a limit on an appropriate Grassmanian, so infinite gains constitute well-defined mathematical objects. The special cases of minimum-order stable and zeroth-order limits are studied in depth, including an analysis of solution behavior. Finally, the classical "cheap control" problem is interpreted within the context of our results.