real analysis - Indeterminate Solutions

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?

Answer

Obviously in real analysis you can not claim that $\frac {1}{0} =\infty$ because as $x\to 0$, $\frac {1}{x}$ takes extremely large positive and extremely large negative (in magnitude) values.

The story of $\frac {1}{\infty}=0$ is a different one. Because as $x\to \infty$, ${1/x}\to 0$

The story of $\frac {0}{0} $ is the most interesting one because that is what we do when we take derivative of a function at a point.

We divide $f(x+h)-f(x)$ by $h$ and let $h\to 0$ so we are trying to find the limit of the difference quotient as both top and bottom tend $0$

As you know not every derivative is $1$ so saying that in advanced courses we may define $\frac {0}{0}=1$ does not make sense.

Word of Wisdom:

Listen to your professors and do not believe the video stuff.