  Asymptotic properties of solutions of difference equation of the form \[
\Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n \] are studied. We give
sufficient conditions under which all solutions, or all solutions with
polynomial growth, or all nonoscillatory solutions are asymptotically
polynomial. We use a new technique which allows us to control the degree of
approximation.
