I understand that. I think I'm just making this a lot more confusing for myself than it needs to be because I'm worried people aren't going to be able to understand the way I say it. Nevermind, and sorry.
Sorry. I guess the only definition of a limit we know of is we can bring a function infinitely close to a limit by choosing x-values infinitely close to the x-value we're approaching.
By "nearly the same number" I'm referring to choosing x-values from the function approaching x from the left...
In the past we would either just know the limit exists due to continuity of a function, and thus being able to substitute x into the function, or from substituting in numbers slightly greater than and less than the x-value, each getting subsequently smaller, until they are nearly the same number.
Ah, sorry, I probably should have specified a bit more. I'm currently working on a project where I have to teach this method to the class, and I'm worried about someone asking why this method is being taught when we have other options. So I'm wondering specifically the advantages of...
I understand the concept of Epsilon-Delta proofs, but I can't understand why we have to do them.
What's the advantage of using this proof over just showing that the limit from the function approaches from the left and right are the same?