Let \(\{V_n\}\) be a sequence of vector spaces with \(V_{n + 1} \subseteq V_n\) for \(n = 0,1,2, \dots\). A series \(\sum^\infty_{n = 0} v_n\) with \(v_n \in V_n\) is called a pre-asymptotic expansion of \(v \in V_0\), written \(v \sim \sum^\infty_{n = 0} v_n\) with respect to \(\{V_n\}\), if \(v - \sum^N_{n = 0} v_n \in V_{N + 1}\) for all \(N\). Under suitable assumptions it is shown that the solution \(v\) of a linear operator equation \(Tv = w\) with \(v \in V_0\), \(w \in \cap^\infty_{n = 0} V_n\) possesses a pre-asymptotic expansion. In particular, for linear differential operators with polynomial coefficients, formal solutions of the form \(\sum_{n = 0}^\infty a_n \delta^{(n)} (x)\) can be interpreted as pre-asymptotic expansions of distributional solutions. Sometimes, pre-asymptotic expansions \(\psi (x) \sim \sum^\infty_{n = 0} \psi_n (x)\) are connected with ordinary asymptotic expansions \(\psi (Ex) \sim \sum^\infty_{n = 0} \psi_n (Ex)\) in the sense of \(\psi (Ex) = \sum^N_{n = 0} \psi_n (Ex) + o(E^N)\) for \(E \to 0\).