Let S be a scheme. Assume that we are given an action of the one dimensional split torus G m , S on a smooth affine S-scheme X . We consider the limit (also called attractor) subfunctor X λ consisting of points whose orbit under the given action ‘admits a limit at 0’. We show that X λ is representable by a smooth closed subscheme of X . This result generalizes a theorem of Conrad et al. (Pseudo-reductive groups (2010) Cambridge Univ. Press) where the case when X is an affine smooth group and G m , S acts as a group automorphisms of X is considered. It also occurs as a special case of a recent result by Drinfeld on the action of G m , S on algebraic spaces (Proposition 1.4.20 of Drinfeld V, On algebraic spaces with an action of G m , preprint 2013) in case S is of finite type over a field.