On equivalence of unbounded metric spaces at infinity

Let $(X,d)$ be an unbounded metric space. To investigate the asymptotic behavior of $(X,d)$ at infinity, one can consider a sequence of rescaling metric spaces $(X,\frac{1}{r_n}d)$ generated by given sequence $(r_n)_{n \in \mathbb N}$ofpositiverealswith$r_n \to \infty$. Metric spaces that are limit points of the sequence $(X,\frac{1}{r_n}d{)}_{n\in \mathbb{N}}$ will be called pretangent spaces to $(X,d)$ at infinity. We found the necessary and sufficient conditions under which two given unbounded subspaces of $(X,d)$ have the same pretangent spaces at infinity. In the case when $(X,d)$ is the real line with Euclidean metric, we also describe all unbounded subspaces of $(X,d)$ isometric to their pretangent spaces.