Well, I would say no, because in some problems, it matters a great deal how the functions behave near the limit, not just what the limiting value is. So here's a proposed counterexample:
which does not exist.
This may or may not be a counterexample. It depends on what you mean by the phrase "consisting [of] f or g". I've got an example that has f and g in it, and I've shown that you can't just swap them out and expect to get the same thing.