Preface: I'm a calculus student who's a bit frustrated because limits seem like the foundation of calculus and seems full of contradiction. If you could answer my questions & address any misconceptions I have, I would be very grateful. Thank you.
So multiple times I've heard the first thing you do want to do to derive the limit of a function is to plug it in.
This drives me crazy since the definition of a limit is what happens around the point not at it but I understand this works for continuous functions e.g. algebraic & transcendental functions are continuous everywhere they are defined.
However sometimes when we plug in values to the function we get an indeterminate form of "0/0" or "inf./inf." or any of the other 6 indeterminate forms. What does that exactly mean? Does this mean that the limit definitely exists & we could find it by algebraic manipulation?
I know that when we find limits and we get the undefined form of "constant/0" that the limit doesn't exist, but are there any other undefined forms in calculus when we know the limit doesn't exist?