Yeah, the limit view is only defined in specific cases. For instance, if you have ((x+3)(x-5))/(x-5), this expression evaluates at x=5 to 0/0, but approaching i

Yeah, the limit view is only defined in specific cases. For instance, if you have ((x+3)(x-5))/(x-5), this expression evaluates at x=5 to 0/0, but approaching it from either side, evaluates to 8. However, (x+3)/(x-5) at x=5 will be approaching -inf on one side and inf on the other. I have an anecdote here too. When I learned about automatic differentiation, I very excitedly told one of my formalist math professors about the quantity e, e^2=0, e!=0, and how it could be used to compute the derivative of a function. He couldn't understand and essentially asked me to motivate the number and sketch the proofs that it's a well-behaved extension, which I couldn't do at all at the time, and he dismissed the whole thing which is actually the basis for some pretty important machine learning these days, among other things. So sometimes you can do pretty weird operations to get an extension, but without an intuitive grasp of how it works - either with a proof it is a well -behaved extension or by analogy for example, it's really hard to tell how useful such a concept is to a mathematician, how well it plays by the rules. Being mathematically rigorous, you will have to define what the views you are talking about. In the one input function limit case, you need to check the limit from both sides. In the 2 input case, you need to check all possible approach paths to prove the limit exists. It gets out of hand. However, these are different views within a formal mathematical system. We don't need to reach consensus among different formal systems which happen to use the same symbols. For instance, I can define the set {cow,0} and equip it with operator '/' such that cow/0= and 0/cow=0 etc. This might be isomorphic to Z mod 2. Then cow by zero is defined, yet we can still agree that division by 0 is undefined, because when we are in general talking about division by 0, we're talking about an operation in a relatively specific context - how numbers behave. So, there really can be a very wide set of perspectives, and being rigorous must also specify the perspectives that count. Also - the cow case is pretty trivial, though this point could also be made with more complex structures, like wheels, that may also have useful physical interpretations, and they still wouldn't affect the collective agreement here because it's a contextually isolated case.