The idea is that what 1/x is is intuitively simple enough, so we should have some standard terminology for it. And saying left and right limits is -infinity or

The idea is that what 1/x is is intuitively simple enough, so we should have some standard terminology for it. And saying left and right limits is -infinity or +infinity really also isn't that weird. I'm pretty sure other metric spaces can be likewise extended and end up with similar algebraic laws as the "extended real numbers". Again this isn't very profound, but is good for education and efficient communication, and so should be perused. Finally, it's interesting that measure spaces with infinite measure is already a thing that people. I would like to see more connections with metric and measure spaces; e.g. we can have an n-point metric which is defined using the measure of the (n-1)-simplex. Just as regular metric spaces have a "triangle inequality", 3-point metric spaces should have a "tetrahedron inequality", and n-point metric spaces should have a "(n-1)-simplex inequality". Again, this is not profound, but good for communication, and connections between definitions help one compress everything for better mental storage.