Thank you, I was actually taking notes on a Topology article from Quanta Magazine a few weeks ago - I intend to make a full-depth Topology page after Multivariable and Differential Equations have been completely fleshed out.
Looking at your properties of limits list, I never understood why they never just say that the function of limits commutes/distributes. It is after all knocked around as if it were an algebraic function.
Left/right hand derivatives sounds like a really weird technicality. I was always under the impression that differentiation was based entirely on continuity determined by the limits. What would we use left/right hand derivatives for?
Left and right hand derivatives are useful to take for a few individual circumstances - Reciprocal Substitution, right after Rule 118, for example. Further uses are will be further elucidated as I publish more rules.