I'm unclear about the use of the phrases infinite language or finite language in computer theory.I think the root of the trouble is that a language like $L=\{ab\}^*$ is infinite in the sense that it can generate an infinite (but countable) number of strings. Yet, it can still be recognized by a finite state automaton. It also doesn't help that the Sipser book doesn't really make this distinction (at least as far as I can tell). A question about infinite/finite languages and their relationship to regular languages came up in a sample exam.
Infinite Language vs. finite language
formal languages;terminology;regular languages