What does the epsilon-delta limit definition ACTUALLY mean ?

[Moamen]

1- How does the epsilon-delta limit definition help us prove that a limit exists ?

I'm not sure how this precise definition agrees with the informal definition ( You can get f(x) as close as you desire to L, provided you get x sufficiently close to a). I mean, if we can show that

| f(x) - L | < epsilon IF 0 < | x - a | < delta

What are we actually doing ? Does this show that the function in question has a definite direction, and that no matter how far we are from L, we will always get there if x gets closer and closer ?

If my understanding is correct, then

2- Why does some people claim that limits only work when x is very close to a?

Isn't the epsilon-delta definition of limits showing that, it doesn't matter how close (x) is to (a) , we can always show that a limit exists if we can provide a delta for a given epsilon, without being restricted to only small deltas?

 

[pka]

1- How does the epsilon-delta limit definition help us prove that a limit exists ?
I'm not sure how this precise definition agrees with the informal definition ( You can get f(x) as close as you desire to L, provided you get x sufficiently close to a). I mean, if we can show that
| f(x) - L | < epsilon IF 0 < | x - a | < delta
What are we actually doing ? Does this show that the function in question has a definite direction, and that no matter how far we are from L, we will always get there if x gets closer and closer ?
If my understanding is correct, then

2- Why does some people claim that limits only work when x is very close to a?
Isn't the epsilon-delta definition of limits showing that, it doesn't matter how close (x) is to (a) , we can always show that a limit exists if we can provide a delta for a given epsilon, without being restricted to only small deltas?

It seems to me that you could use a really rigorous visual explication.
Here is a calculus textbook that does that. It is a free download.
In Elementary Calculus: An Infinitesimal Approach
An Infinitesimal Approach by Jerome Keisler.
Download chapters #1,2,&, 3.

 

[Prove It]

Think of yourself as working in a pizza shop. When the pizzas go in the oven, everybody has a picture of the perfectly cooked pizza. We could call this level of cooking "the limit". This level of cooking has a certain time associated to it.

Obviously it is going to be impossible to have the pizzas be in the oven for this exact ideal amount of time 100% of the time (it might not even be possible to do at all). But we each allow a certain TOLERANCE before we consider the pizza overcooked or undercooked.

The idea is, that as long as you are SUFFICIENTLY CLOSE to the time required for the ideal level of cooking, then you will be in this tolerance.

Also, as you gain experience, you are likely to get better at picking the right amount of time, and thus are going to decrease your tolerance. So your tolerance should decrease as you close in on the correct amount of time. This means that the amount of time is related to the tolerance.

That is all the precise definition of a limit is. If we set \(\displaystyle \displaystyle \begin{align*} x \end{align*}\) to represent the amount of time in the oven, then \(\displaystyle \displaystyle \begin{align*} f(x) \end{align*}\) represents the level of cooking.

We set \(\displaystyle \displaystyle \begin{align*} L \end{align*}\) to represent the ideal level of cooking, and \(\displaystyle \displaystyle \begin{align*} \epsilon \end{align*}\) to represent our tolerance (notice that this value would be freely chosen).

We can use \(\displaystyle \displaystyle \begin{align*} a \end{align*}\) to represent the time associated with the ideal level of cooking, and \(\displaystyle \displaystyle \begin{align*} \delta \end{align*}\) to represent the amount of leeway in time.

So going back to our original statement, we choose our cooking tolerance, some \(\displaystyle \displaystyle \begin{align*} \epsilon > 0 \end{align*}\), and as long as you have the pizzas in the oven for an amount of time that is sufficiently close to the ideal time - in other words, if the absolute difference between the actual time and our ideal time is within a sufficiently close leeway, so \(\displaystyle \displaystyle \begin{align*} 0 < |x - a| < \delta \end{align*}\) (we have to put 0 there because there's no guarantee it actually can be in there for the ideal amount of time), then we are certain that our pizza will have a level of cooking within our tolerance, in other words, \(\displaystyle \displaystyle \begin{align*} \left| f(x) - L \right| < \epsilon \end{align*}\). And since the level of cooking is related to the time in the oven, then \(\displaystyle \displaystyle \begin{align*} \delta \end{align*}\) must also be related to \(\displaystyle \displaystyle \begin{align*} \epsilon \end{align*}\).

So putting it all together, when we think of a limit as an "ideal value" that gets squeezed in on, then \(\displaystyle \displaystyle \begin{align*} \lim_{x \to a} f(x) = L \end{align*}\) if we can show that for all \(\displaystyle \displaystyle \begin{align*} \epsilon > 0 \end{align*}\) that there exists a value \(\displaystyle \displaystyle \begin{align*} \delta >0 \end{align*}\) which is a function of \(\displaystyle \displaystyle \begin{align*} \epsilon \end{align*}\) such that \(\displaystyle \displaystyle \begin{align*} 0 < |x - a| < \delta \implies \left| f(x) - L \right| < \epsilon \end{align*}\).

 

[Steven G]

1- How does the epsilon-delta limit definition help us prove that a limit exists ?

I'm not sure how this precise definition agrees with the informal definition ( You can get f(x) as close as you desire to L, provided you get x sufficiently close to a). I mean, if we can show that

Given epilson>o we can find delta>o such that

| f(x) - L | < epsilon IF 0 < | x - a | < delta

What are we actually doing ? Does this show that the function in question has a definite direction, and that no matter how far we are from L, we will always get there if x gets closer and closer ?

If my understanding is correct, then

2- Why does some people claim that limits only work when x is very close to a?

Isn't the epsilon-delta definition of limits showing that, it doesn't matter how close (x) is to (a) , we can always show that a limit exists if we can provide a delta for a given epsilon, without being restricted to only small deltas?

No! As pka pointed out you need to see a visual diagram of this. It is good that you came here to ask this question! You now need to convince your self that the definition is correct. If you still do not see this then please come back.

 

[Moamen]

Thank you all, that was very helpful. I just have another question,

Prove it said in his very exciting pizza story that,

The idea is, that as long as you are SUFFICIENTLY CLOSE to the time required for the ideal level of cooking, then you will be in this tolerance.

Does that mean that epsilon can be as big or small as we like, but delta has to be small enough so that all of the X's within its range will be sent to f(x)'s within epsilon range?

In other words, delta doesn't have to provide x's for all of the f(x)'s within epsilon in order for the limit to exist ?

 

[stapel]

Prove it said in his very exciting pizza story that, "The idea is, that as long as you are SUFFICIENTLY CLOSE to the time required for the ideal level of cooking, then you will be in this tolerance."

Does that mean that epsilon can be as big or small as we like, but delta has to be small enough so that all of the X's within its range will be sent to f(x)'s within epsilon range?

Yes. The idea is that you'll give me the closeness to the limit or output value that you want (in other words, the tolerance or epsilon that you want your y-value to be within), and I'll give you a neighborhood of input values (in other words, a delta-based range of x-values) that will guarantee that you're at least that close. You'll tell me how close you want to be to the target when you land, and I'll tell you where to stand before you jump.

In other words, delta doesn't have to provide x's for all of the f(x)'s within epsilon in order for the limit to exist ?

I'm not sure what you mean by this...?

 

[Moamen]

Yes. The idea is that you'll give me the closeness to the limit or output value that you want (in other words, the tolerance or epsilon that you want your y-value to be within), and I'll give you a neighborhood of input values (in other words, a delta-based range of x-values) that will guarantee that you're at least that close. You'll tell me how close you want to be to the target when you land, and I'll tell you where to stand before you jump.


I'm not sure what you mean by this...?

I just realized, while trying to rephrase my question, that I have a SERIOUS misunderstanding of Limits. I'll go over the concept of limits again and hopefully come back with at least understandable questions.