Information-theoretic results indicate that multi-user multi-antenna systems provide potentially huge capacity gains. However, these results assume sophisticated coding techniques and perfect channel state information which may require a prohibitive amount of feedback. We show that transmitting to just a few suitably selected users is asymptotically optimal as the user population grows large, while dramatically reducing the feedback overhead and operational complexity. Specifically, we consider a multi-antenna broadcast system with M transmit antennas, and a heterogeneous user population. we tackle the problem of maximizing a weighted sum rate quantity. We establish a novel upper bound for the weighted sum capacity, which we then use to show that the maximum expected weighted sum rate can be asymptotically achieved by transmitting to a suitably selected subset of at most MC users, where C denotes the number of distinct user classes. Numerical experiments indicate that the asymptotic results are remarkably accurate and that the proposed schemes operate close to absolute performance bounds, even for a moderate number of users.