  The problem of extrapolating asymptotic perturbation-theory expansions in
powers of a small variable to large values of the variable tending to infinity
is investigated. The analysis is based on self-similar approximation theory.
Several types of self-similar approximants are considered and their use in
different problems of applied mathematics is illustrated. Self-similar
approximants are shown to constitute a powerful tool for extrapolating
asymptotic expansions of different natures.
