A graph G is said to be 2-distinguishable if there is a 2-labeling of its vertices which is not preserved by any nontrivial automorphism of G. We show that every locally finte graph with infinite nite motion and growth at most o(2((1-epsilon)/root n/2)) is 2-distinguishable. Infinite motion means that every automorphism moves infinitely many vertices and growth refers to the cardinality of balls of radius n.