Kragujevac J. Math. 24 (2002) 71-79.

1. In [1] S. Banach solved the problem of the existence of a (non-unique) linear shift-invariant functional on the space of all bounded functions defined on the semi-axis t ³ 0.

2. Let now a be sufficiently large (written a > a₀ for some a₀). Denote by W the real vector space of all real-valued functions on [0,¥) and bounded on [a,¥). This paper is organized as follows. First we will show the existence of a family of functionals on the space W containing Banach shift-invariant functionals. Next, by these functionals we shall define the limit of f(t) as t®¥, f Î W, and show that this definition is equivalent to the classical definition of this limit. Further, we show some theorems characterizing the limit of a function f(t), t ³ 0 as t®+¥. Each of these theorems gives an answer to the question what (new) conditions must satisfy a function f Î W such that the limit of f(t) as t®+¥ exists.