In this paper, for a set $X,$ a scalar valued function
$f:X\rightarrow\Bbb F,$ where $\Bbb F$ is assumed to be the set of
all real or complex numbers, and for a cardinal number $\alpha,$
we define the $\alpha$-limit of $f$ as a generalization of the
limit of a sequence
and denote it by $\lim^{\alpha} f.$
Indeed, if we assume that $\alpha=\aleph_0$ and $X=\Bbb N,$ then
$\lim^{\alpha} f$ is equal to $\lim_{n\to \infty}f(n).$ If $X$ is an
infinite set and $\alpha$ an infinite cardinal number, then we
define two classes of Banach spaces ${\mathfrak c}_0^{\alpha}(X)$
and ${\mathfrak c}^{\alpha}(X)$ containing all bounded functions
$f:X \to \Bbb F$ which satisfy $\lim^{\alpha} f =0$ and that
$\lim^{\alpha} f$ exists, respectively. Then we show that
${\mathfrak c}_0(X)\subseteq {\mathfrak c}_0^{\alpha}(X)\subseteq
\ell^{\infty}(X)$ and ${\mathfrak c}(X)\subseteq {\mathfrak
c}^{\alpha}(X)\subseteq \ell^{\infty}(X).$ Also, whenever the
cardinal number $\alpha$ increases
these spaces become larger such that the equality hold on the
left sides for $\alpha =\aleph_0$ and on the right sides for all
$\alpha>\card(X).$
