  The main goal of this article is to put some known results in a common
perspective and to simplify their proofs.
  We start with a simple proof of a result of Vereshchagin saying that
$\limsup_n C(x|n)$ equals $C^{0'}(x)$. Then we use the same argument to prove
similar results for prefix complexity, a priori probability on binary tree, to
prove Conidis' theorem about limits of effectively open sets, and also to
improve the results of Muchnik about limit frequencies. As a by-product, we get
a criterion of 2-randomness proved by Miller: a sequence $X$ is 2-random if and
only if there exists $c$ such that any prefix $x$ of $X$ is a prefix of some
string $y$ such that $C(y)\ge |y|-c$. (In the 1960ies this property was
suggested in Kolmogorov as one of possible randomness definitions.) We also get
another 2-randomness criterion by Miller and Nies: $X$ is 2-random if and only
if $C(x)\ge |x|-c$ for some $c$ and infinitely many prefixes $x$ of $X$.
  This is a modified version of our old paper that contained a weaker (and
cumbersome) version of Conidis' result, and the proof used low basis theorem
(in quite a strange way). The full version was formulated there as a
conjecture. This conjecture was later proved by Conidis. Bruno Bauwens
(personal communication) noted that the proof can be obtained also by a simple
modification of our original argument, and we reproduce Bauwens' argument with
his permission.
