  We prove new results on upper and lower limits of real-valued functions by
means of $\psi$-densities introduced by P. D. Barry in 1962. This allows us to
improve several existing results on the growth of non-decreasing and unbounded
real-valued functions in sets of positive density. The $\psi$-densities are
also used to introduce a new concept of a limit for real-valued functions. The
results in this paper are of interest in real analysis as well as in the theory
of meromorphic functions.
