Let $\{A_n\}$ be an inverse system of rings and $\hat{A}$ be the inverse limit of this system. Let $K_2(R)$ denote Milnor $K_2$ (I will assume that the case I am interested in the Milnor $K_2$ is isomorphic to algebraic $K_2$ of Quillen).
Question: Is $K_2(\hat{A}) \cong {\rm inverse~ lim}\ K_2(A_n)$?
I am interested in an answer to this question in reasonably "nice" situations. For ex. take $R=k[x_1,\cdots,x_n]$ and let $I$ be any smooth, complete intersection ideal (for instance $I=(f)$ for some smooth, irreducible polynomial). Then let $A_n=R/I^n$ and $\hat{A}$ is the completion.