  Let $(X,d)$ be an unbounded metric space and $\tilde r=(r_n)_{n\in\mathbb N}$
be a scaling sequence of positive real numbers tending to infinity. We define
the pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity as
a metric space whose points are equivalence classes of sequences
$(x_n)_{n\in\mathbb N}\subset X$ which tend to infinity with the speed of
$\tilde r$. It is proved that the pretangent spaces $\Omega_{\infty, \tilde
r}^{X}$ are complete for every unbounded metric space $(X, d)$ and every
scaling sequence $\tilde r$. The finiteness conditions of $\Omega_{\infty,
\tilde r}^{X}$ are found.
