We consider the result of a wet splicing procedure after the reaction has run to its completion, or limit, and we try to describe the molecules that will be present at this final stage. In language theoretic terms the splicing procedure is modeled as an H system, and the molecules that we want to consider correspond to a subset of the splicing language which we call the limit language. We give a number of examples, including one based on differential equations, and we propose a definition for the limit language. With this definition we prove that a language is regular if and only if it is the limit language of a reflexive and symmetric splicing system.