Limit points in the range of the commuting probability function on finite groups

If G is a finite group, then Pr(G) denotes the fraction of ordered pairs of elements of G which commute. We show that if l is an element of (2/9, 1] is a limit point of the function Pr on finite groups, then l is an element of Q and there exists an epsilon = epsilon(l) > 0 such that Pr(G) (sic) (l -epsilon(l), l) for any finite group G. These results lend support to some old conjectures of Keith Joseph.