I Why use Epsilon Delta proofs?

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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?
 

RPinPA

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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.
 
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.
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!
 

PeroK

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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!
What's your alternative definition of a limit; or of continuity of a function?
 

fresh_42

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Limits are just an abbreviation. If you resolve them by their meaning, you will end up with a epsilon-delta version of continuity, It are equivalent definitions, so it's a matter of taste or context which one fits better.
 
What's your alternative definition of a limit; or of continuity of a function?
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.
 

PeroK

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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.
That's not a definition of anything. What do you even mean by "nearly the same number"?
 
That's not a definition of anything. What do you even mean by "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.

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 (nearly) the same value.
 

fresh_42

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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.
... and if you formally make this rigorous you end up with ##\varepsilon-\delta##.
 
... and if you formally make this rigorous you end up with ##\varepsilon-\delta##.
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.
 

PeroK

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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.
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.
 

symbolipoint

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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?
What I say here is not really an answer, but I feel some explanation is needed anyway.
You need to be comfortable with and can understand these epsilon-delta proofs yourself if you want to teach them for derivatives, assuming you will teach semester 1, Calculus 1, to high school or college students, and you yourself should probably have at least the equivalent of undergraduate degree in Mathematics. Use of variables to PROVE limits of functions is part of studying Calculus. These proofs must be there because just waving numbers (values) around is not enough to PROVE. Your students might have too much trouble understanding but YOU must not have too much trouble understanding nor presenting. Maybe you will need to restudy this stuff on your own before you try to teach it. Maybe you can do your own review from similar books which your students will use.
 
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The ε-δ concept is worth learning itself, regardless of what the specific proof is. It is a concept fundamental to calculus. If you students don't understand it, then they won't really understand calculus, IMO; even if they can still solve homework problems.
 

statdad

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"we can bring a function infinitely close to a limit by choosing x-values infinitely close to the x-value we're approaching.

The words I've highlighted there are weasel words -- your phrasing has no meaning since you don't specify what you mean by "infinitely close".

"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."

More weasel words at "but choose small enough values" -- how do you know when you're close enough? How do you know that the cutoffs you choose for "close" give the correct answer for what occurs when you're even closer?

I think the very nice behavior of the functions you (and, to be honest, everyone early in their math training) are familiar with give a false indication of the behaviors of functions in general: they're too easy to picture, so your
intuitive message about what limits are "works" for them. It won't help for more complicated functions.

A couple other points:

  • this "get close" idea for limits would need a good bit of wording adjustment to deal with limits at infinity -- wording that I don't believe would be easy to motivate in a logical sense. The traditional limit definition a point is easy to modify
  • For any students that go on to more advanced settings the ## \varepsilon## and ##\delta## idea
  • will be crucial, whether you are dealing with limits relating to integration, functions of several variables, or more abstract notions. Postponing the introduction of the notation and the familiarity of working with it will only hamper those students' progress
By all means relate the ##\varepsilon## - ##\delta## idea to the intuitive ideas, but introduce the formal definition first.
 

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