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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 definethe pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity asa 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 $\Omega_{\infty, \tilder}^{X}$ are complete for every unbounded metric space $(X, d)$ and everyscaling sequence $\tilde r$. The finiteness conditions of $\Omega_{\infty,\tilde r}^{X}$ are found.

Asymptotic structure of general metric spaces at infinity