We will see there is a well defined limit, in theGrothendieck ring of varieties, of several different sequencesofmoduli spaces. In flavor, these are algebro-geometric analogs of topological results about homologicalstability, or of arithmetic results about asymptotics of points countson moduli spaces. For example, we determine the limit of the motiveof smooth divisors (or with s singularities) inincreasing multiplesof a linear system. We also determine the limit of the motive ofconfiguration spaces of distinct points (or points that are allowed tocome together to a limited extent) as the number of points increases.All of these limits in the Grothendieck ring are given by explicitformulas in terms of motivic zeta values. Our results motivate a largenumber of conjectures in topology and arithmetic. This is joint workwith Ravi Vakil.