The ℏ→0 limit of the quantum dynamics determined by the Hamiltonian H(ℏ) =−(ℏ²/2m)Δ+V on L²(Rⁿ) is studied for a large class of potentials. By convolving with certain Gaussian states, classically determined asymptotic behavior of the quantum evolution of states of compact support is obtained. For initial states of class C¹/₀ the error terms are shown to have L² norms of order ℏ^1/2−ε for arbitrarily small positive ε.