The answer is "no". The lamplighter group (which is infinitely presented) is a limit of a sequence of virtually free groups and surjective homomorphisms (see, for example, this [question and answers there][1]). All virtually free groups are residually finite.  


  [1]: https://mathoverflow.net/questions/75784/is-there-a-non-hopfian-lacunary-hyperbolic-group