How to choose a limit for the epsilon delta proof?

Question

The epsilon delta proof allows us to prove that a 'given' limit is actually the limit. In the sense that, we can show using that if a limit for a sequence/ function exist, then that limit is a unique one. But, how does one find a limit following the epsilon delta definition or do we need something more?

Answer 1

It depends on the limit. Sometimes, it's obvious and you can just eyeball it. Sometimes, you have to multiply and divide by the relevant quantity in order for it to become obvious. Sometimes, you have to just look at the graph to get a sense of what the limit is.

Sometimes, you actually may have to apply formulas or theorems which are not necessarily applicable just to have an answer to work with. Keep in mind that theorems have conditions and if the conditions are not satisfied by whatever it is that you're dealing with, then you cannot use that theorem. You either have to do something else or you have to modify the thing you're dealing with sufficiently to apply the theorem.

But there's no law that tells you that you can't just sneakily cheat and apply a theorem where it may not be applicable just to get a sense of what the limit is. There's no law that's telling you that you can't just approximate functions by their Taylor expansions and just use that to get a rough estimate for what the limit is.

All of this can be done on scrap paper and, so to speak, is hidden away from public view. We don't publicly tell everyone that we found a limit by L'Hopital's rule. We just tell them what the limit is and prove that it is the limit via the golden standards of proof-writing, whether it be by an $\epsilon-\delta$ argument or by making use of other theorems (by the way, making use of theorems to prove a limit where it is justified is perfectly rigorous.)