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Limit spaces of a discrete metric space capture the local geometry as one 'walks off to infinity in a given direction': for a homogeneous space like a group, they are just copies of the original space, but in general they can be quite different. To a band-dominated operator on a discrete space X, there are families of limit operators defined on the associated limit spaces that govern aspects of the 'behavior at infinity' of the operator (for example, whether it is Fredholm). I'll explain how these constructions work, and the relevance of Yu's property A (a large-scale version of amenability) for understanding this. Some of this will be based on joint work with Jan Spakula.