Asymptotic growth of multiplicity functions

We consider several multiplicity functions associated with a pair of ideals J⊆IJ⊆I in a local noetherian ring R. In particular, given an arbitrary ideal J   and an element x∈Rx∈R, we show that for each m   the multiplicity f(n)f(n) of the R  -module (J+xR)n/(J+xR)n−mJm(J+xR)n/(J+xR)n−mJm is eventually a constant φJ,x(m)φJ,x(m) which is non-zero only in the case when x is not integral over J  . We study the asymptotic growth of φJ,x(m)φJ,x(m) and some other multiplicity functions.