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Numerical calculations made by several groups have always shown that an asymptotic lower exit emittance exists for linacs operating with high beam current as the input emittance is reduced to zero. In this paper, a mechanism for this limit is shown to be spread in the betatron frequencies of the individual particles due to the combination of space charge and r.f. gap forces, causing different transverse trajectories correlated with the instaneous longitudinal position of the particle. These trajectories cannot all be simultaneously matched to the average restoring forces, resulting in an overall emittance increase if the space charge force is a large fraction of the restoring force. In principle, equilibrium distributions may exist which would not grow. Raising the linac frequency or reducing longitudinal emittance improves the situation, but higher injection energy does not.