We introduce a new computational model for data streams: asymptotically exact streaming algorithms. These algorithms have an approximation ratio that tends to one as the length of the stream goes to infinity while the memory used by the algorithm is restricted to polylog(n) size. Thus, the output of the algorithm is optimal in the limit. We show positive results in our model for a series of important problems that have been discussed in the streaming literature. These include computing the frequency moments, clustering problems and least squares regression. Our results also include lower bounds for problems, which have streaming algorithms in the ordinary setting but do not allow for sublinear space algorithms in our model.