1. In [1] S. Banach shown the existence of very known Banach linear shift-invariant functionals defined on the real vector space of all bounded real-valued functions on the semi-axis <em>t</em>≥0 and especially on the space of all real bounded sequences. In [2] G.G. Lorentz defined, by Banach shift-invariant functionals, the class of almost convergent sequences. In [3] almost convergence was extended to real-valued functions on the semi-axis $t\geq 0$. In [4] almost convergence was extended to bounded sequences in a real normed space. 2. This paper is devoted to a class of functions defined on the semi-axis $t\geq 0$, which are near to the functions $f$ having $lim_{t\to \infty}f(t)$. The paper is organized as follows. First, for a sufficiently large $a$ (written $a > a_0$ for some $a_0$) by $\Omega$ we denote the real vector space of all functions defined on $[0,+\infty)$. Next, we will show the existence of a family of functionals defined on the space $\Omega$. By these functionals we define the notion of a function $f\in\Omega$ and investigate the family of all these functions. Further, we will show a theorem characterizing a function having a pre-limit. Also, we show another theorem which is very applicable, though it contains a new restrictive condition. Finally, to make the idea of pre-limit a little clearer, we give several examples functions having pre-limit.