So, the 00 in the definition "0<|x−y|0<|x−y|" is of critical importance because it is the subtle point which separates taking a limit from just doing algebra. Although, I will admit, the majority of the limits we encounter in first semester calculus are just that; algebra. But...
I know I have had the experience of trying to ask this on mathstackexchange and getting a bunch of answers telling me to learn homological algebra and Ext, which is more effort than a poor undergraduate wants to spend to get at a concrete problem. However, I think there is a natural dir...