I know a couple of different ways to prove a limit like the $\epsilon-\delta$ approach and squeeze theorem. I know you can find a limit using squeeze theorem, definition of derivative, l'hospital, taylor series, and some algebraic or trigonometric approach.I know finding a limit algebraically does not count as a proof...However some seem to think finding a limit the l'hospital way is proof. If so why? Why is l'hospital proof while the algebraic way is not? Also please add any other ways to my list above to prove limits if you can think of any other general ways. Thanks kindly.
It depends what you mean. An $\epsilon-\delta$ proof follows directly from the formal definition of a limit, however evaluating limits in other means obviously works and is much easier than using an $\epsilon-\delta$ argument every time. These methods have been proved to work so they are valid ways of "proving" limits however in a real-analysis class they usually want you to prove everything from the axioms, so for example you can prove a limit using L'Hopital's rule but you'd have to prove L'Hopital's rule first, in calculus however they just teach you the evaluation methods and you have to just assume that they work. Hope this helps!