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 Epsilon-Delta proofs. (Sorry if I'm not making a lot of sense, I might not have as strong of an understanding as I thought). Thanks!If you have another way of showing it other than applying the definition of limit (the epsilon-delta statement), then that's fine.
For instance, you might be able to take advantage of some limit theorems. But those theorems were most likely originally proved with epsilon-delta arguments.
What's your alternative definition of a limit; or of continuity of a function?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 Epsilon-Delta proofs. (Sorry if I'm not making a lot of sense, I might not have as strong of an understanding as I thought). Thanks!
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.What's your alternative definition of a limit; or of continuity of a function?
That's not a definition of anything. What do you even mean by "nearly the same number"?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.
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.That's not a definition of anything. What do you even mean by "nearly the same number"?
... and if you formally make this rigorous you end up with ##\varepsilon-\delta##.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 and the right, like 2.001 and 1.999 as we approach 2, but choose small enough values that f(x) yields the same value.
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.... and if you formally make this rigorous you end up with ##\varepsilon-\delta##.
I think it's fair to say that unless you are clear in your own mind, then it's unlikely to be clear in the minds of your students.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.
What I say here is not really an answer, but I feel some explanation is needed anyway.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?