I was watching an old video by Mathologer talking about various problems involving 0 and $\infty$, but at the end of the video, at roughly 11:40, he concludes
...if you want to make sense of $\frac30$ you also do this by sneaking up on $0$ and of course you know that things explode magnitude-wise...In higher level calculus it actually makes sense to treat infinity like a number and to actually write equations like $\frac30=\infty$, and you really mean it...$3$ as a number divided by infinity as a number is equal to infinity.
He then goes on to say
In other branches of mathematics, you sometimes find it actually does make sense to set $\frac00$ equal to 1
He said he would make a video elaborating on these last claims, but after digging through his playlists I don't think he has. The things said here sound like mathematical heresy according to what I've been told by math professors. My first guess right now is to not take his exact words at face value, but consider the gist of what he's trying to say is analogous to say, $=$ signs having different meanings in different contexts, i.e., a regularized sum vs assigned sum. Maybe what he's saying really is true. But I want to make sure, considering I don't think he ever made a follow-up.
So, can anybody confirm that he is wrong or that I'm just missing some crucial context?