One of the most important constructions in geometric group theory is the asymptotic cone of a group or metric space, which captures what happens to the group or metric space as you rescale the metric down to zero (or, less formally, as you squint your eyes and move farther and farther away from the space). The originated in Gromov's original proof of his polynomial growth theorem, where the polynomial growth condition assures that this limit exists in the classical sense. However, since then this concept has been hugely important in more general situations where one has to use an ultrafilter to define an appropriate notion of convergence. I recommend perusing the wikipedia articles on Gromov-Hausdorff convergence and ultralimits.

One of the most important constructions in geometric group theory is the asymptotic cone of a group or metric space, which captures what happens to the group or metric space as you rescale the metric down to zero (or, less formally, as you squint your eyes and move farther and farther away from the space). The originated in Gromov's original proof of his polynomial growth theorem, where the polynomial growth condition assures that this limit exists in the classical sense. However, since then this concept has been hugely important in more general situations where one has to use an ultrafilter to define an appropriate notion of convergence. I recommend perusing the wikipedia articles on Gromov-Hausdorff convergence and ultralimits.