Asymptotic prime ideals related to derived functors

Let $R$ be a commutative noetherian ring. Let $N$ (resp. $A$) denote a noetherian (resp. artinian) $R$-module and $I$ an ideal of $\left[ R \right]$. It is shown that for each integer $i$ the sets of prime ideals ${\operatorname{Ass} _R}\operatorname{Tor} _i^R(R/{I^n},N)$ and ${\operatorname{Att} _R}\operatorname{Ext} _R^i(R/{I^n},A),\;n = 1,2, \ldots$, become for $n$ large independent of $n$.