We study languages which can be described as limits of fast converging infinite sequences of context-free languages. Such a sequence \(L_0 \subseteq L_1 \subseteq L_2 \subseteq\) ... is fast converging if each string w of its limit language belongs to an Li which has a grammatical description very concise in comparison with the length of w . We prove that these languages are closely related to context-free languages in several properties: pumping lemma, interchange lemma, regularity of unary languages, full AFL properties. The languages can differ, however, in their computational complexity: we construct languages of arbitrarily high complexity, even languages which are not recursively enumerable, but have fast context-free approximations.