Epsilon-delta definition of a limit

[koolraj09]

Hi guys.
I need to understand the Epsilon-delta definition of a limit both geometrically and physically. Does it have a physical meaning?
Is there really a necessity for this definition?
Thanks.

 

[chiro]

Intuitively it just means that the closer you get distance wise from some point, the smaller the delta gets between the limit and its value at a point.

In a general setting we use something to measure distance: In a euclidean geometry we use the pythagorean norm which is basically SQRT(v.v) where v is the vector from point where limit is defined to other arbitrary point.

So in your one dimensional calculus your norm is simply SQRT([x-a]^2) which is simply |x-a|, In two dimensions its SQRT([x-a]^2 + [y-b]^2) which is just a norm in two dimensions.

So for euclidean geometries use the standard pythagorean norm. For other geometries you use another norm.

The reason we have to use norm is to basically say that the closer we get from any direction to the point in question, the closer we get to the limit. In one dimension we only have one dimension, but in higher dimensions we can approach the limit from many different directions. So the intuitive way to make sure that limits do what they're meant to with these larger degrees of freedom is to use the normed definition where we use differences in distances to correspond to distances in deltas.

 

[Rasalhague]

Hi, koolraj09, it does seem quite a complicated and forbidding definition at first sight, but it's useful in analysis for making rigorous statements about limits. Once you get the hang of it, you might find it helpful to think about ways of paraphrasing the definition that sound more intuitive to you.

There's a nice, gentle introduction to the idea in these two videos from the Khan Academy:

http://www.khanacademy.org/video/epsilon-delta-limit-definition-1?playlist=Calculus

http://www.khanacademy.org/video/epsilon-delta-limit-definition-1?playlist=Calculus

*

Suppose we have a real-valued function of real numbers, $f: \mathbb{R} \rightarrow \mathbb{R}$. (That means the input of $f$ is a real number, and so is the output.) To call a real number, $L$, the limit of $f(x)$ (the outputs) as the independent variable $x$ (the input) approaches a specific real number $x_0$ is to say this:

"Think of a positive real number, any positive real number, $\epsilon$, representing a distance of $\epsilon$ units away from $L$. No matter what distance, $\epsilon$, you thought of, however small, I can always give you at least one positive real number, $\delta$ (which could be different for different epsilons), representing a distance of $\delta$ units away from $x_0$, such that the following statement is true.

If the input, $x$, is less than distance $\delta$ away from $x_0$, then the output, $f(x)$ will be less than distance $\epsilon$ away from $L$."

In other words, for $L$ to be the limit, the stuff in bold has to be true for every real number $\epsilon$.

*

More formally, for $L$ to be the limit of $f(x)$ as $x$ approaches $x_0$ means that for every real number, $\epsilon > 0$, there exists a real number, $\delta > 0$, such that if $|x - x_0| < | \delta - x_0 |$, then $|f(x) - L| < \epsilon$.

In that case, we write

$$\lim_{x\rightarrow x_0} f(x) = L.$$

*

By choosing an appropriate definition of distance, as Chiro discusses, the idea of the epsilon-delta limit can also be applied to Euclidean spaces with more than one dimension, and to other metric spaces. But you don't need to know about these generalisations to understand it in one dimension.

 

[Edgardo]

Recognize the similarity to the limit-definition of a sequence.

We say a sequence b_n is convergent to the limit b if
$$\forall \epsilon > 0 \; \exists n_{0} \in \mathbb{N}: |b_{n} - b| < \epsilon \quad \forall n>n_0$$
Or intuitively: If you give me an epsilon I can show you that the distance of b_n to b will become less than epsilon if I make n big enough.
Or simpler: The bigger you make n the more b_n will approach b.

The epsilon-delta definition is similar:
The limit of f(x) as x approaches x₀ is L if
$$\forall \epsilon > 0 \; \exists \delta>0 : ( |x-x_0|<\delta \Rightarrow |f(x)-L|<\epsilon )$$
Or intuitively: If you give me an epsilon I can show you that the distance of f(x) to L will become less than epsilon if I let x get near enough to x₀.
Or simpler: The more you let x get near to x₀ the more f(x) will approach L.


The difference between both definitions is:
In the sequence limit you let n get near to infinity.
In the function limit you let x get near to x₀.

 

[arildno]

Hi, kooolraj!

1. Draw any particular graph, discontinuous or continuous.

2. Now, draw a curve(+e) ABOVE that graph, so that the vertical distance from (+e) to your graph always is some number e(psilon).
3. Draw a similar curve(-e) BELOW your graph.

4. You have now three parallell curves.
Continuity means that for each point on your graph, you may let that point be the centre of a circle with radius d(elta), so that the entire circle is contained within the created "e-tube" (i.e, the space created between (+e) and (-e)).

At a point of discontinuity, this will be impossible if you make the number "e" small enough.
(just make "e" less than than the value jump at the point of discontinuity!)

 

[Excalibur1152]

In a limit, the x-value can never equal the x-coordinate of the limit (c). That means that there will always be a distance between x and "c."

then:

There will always be a distance between y and L.

 

[HallsofIvy]

In a limit, the x-value can never equal the x-coordinate of the limit (c). That means that there will always be a distance between x and "c."

then:

There will always be a distance between y and L.

That does not follow. Yes "the x-value can never equal the x-coordinate of the limit (c)". That comes from the "0<" in $0< |x- c|< \delta$ part of the definition of limit.

But it does NOT follow from that "there will always be a distance between y and L". For example, if y= f(x) is a constant, f(x)= L for all x, then y= L no matter how far x is from c.

 

[arildno]

Aah, yes! My old student day definition of a hypercontinuous function:

A function is hypercontinuous at a point X iff there exists a number L, so that for every delta&epsilon>0 we have:
$$|f(x)-L|<\epsilon$$,
whenever we have $$|x-X|<\delta$$, and otherwise.

Glad you brought back good memories, Halls!

 

[disregardthat]

Aah, yes! My old student day definition of a hypercontinuous function:

A function is hypercontinuous at a point X iff there exists a number L, so that for every delta&epsilon>0 we have:
$$|f(x)-L|<\epsilon$$,
whenever we have $$|x-X|<\delta$$, and otherwise.

Glad you brought back good memories, Halls!

In other words constant ? :)

 

[arildno]

In other words constant ? :)

That equivalence needs to be established by means of a proof, given by Arildno's 2.theorem.

My first theorem concerns Siamese Primes in number theory.
It is a major result, but the study of Siamese Primes have never really caught on among professionals..

 

[disregardthat]

My first theorem concerns Siamese Primes in number theory.
It is a major result, but the study of Siamese Primes have never really caught on among professionals..

Enlighten us, all I could find about this obscure topic was that it was pairs of primes on the form $$(n^2-2,n^2+2)$$.

 

[arildno]

Since Siamese Twins are more closely stuck together than ordinary twins, I thought it was fairly obvious.

To primes "a" and "b" are called siamese (in my usage) iff |a-b|=1.

My theorem concludes that the number of Siamese Primes is non-zero, but finite.

 

[disregardthat]

Since Siamese Twins are more closely stuck together than ordinary twins, I thought it was fairly obvious.

To primes "a" and "b" are called siamese (in my usage) iff |a-b|=1.

My theorem concludes that the number of Siamese Primes is non-zero, but finite.

That's a surely a more interesting definition.

I have a corollary to your theorem: Every infinite subset of the Siamese primes contains exactly six elements.

 

[arildno]

That's a surely a more interesting definition.

I have a corollary to your theorem: Every infinite subset of the Siamese primes contains exactly six elements.

A beautiful corolary!

As I said, the theory of Siamese Primes is woefully under-developed, and many corolarian gems like yours could be unearthed if the professionals had only become interested.
I think it is a conspiracy here..

 

[disregardthat]

I think it is a conspiracy here..

Mathematicians cannot be trusted, and they are only interested in the meaningless blather of which mathematics is on higher levels consist of. They earn their salary by collectively insisting on the importance of their obscure symbolism that no one else understands. Developing what they call theory around a beautiful and understandable subject such as Siamese primes could quite possibly expose their agenda.

 

[arildno]

Mathematicians cannot be trusted, and they are only interested in the meaningless blather of which mathematics is on higher levels consist of. They earn their salary by collectively insisting on the importance of their obscure symbolism that no one else understands. Developing what they call theory around a beautiful and understandable subject such as Siamese primes could quite possibly expose their agenda.

Indeed!
What other set of primes has the fascinating property that given distinct "a", and "b", we have:
$$|a^{b}-b^{a}|=1$$?

There is so much beauty in the set of Siamese primes the world just is longing to be exposed to..

 

[disregardthat]

Indeed!
What other set of primes has the fascinating property that given distinct "a", and "b", we have:
$$|a^{b}-b^{a}|=1$$?

There is so much beauty in the set of Siamese primes the world just is longing to be exposed to..

Excellent, the theory of siamese twins is rapidly expanding.

It has some interesting topological properties as well. Any continuous function from the Siamese primes to the Siamese primes is either a homeomorphism or hypercontinuous on the whole domain under the induced topology as a subspace of the metric space R!

 

[arildno]

Excellent, the theory of siamese twins is rapidly expanding.

It has some interesting topological properties as well. Any continuous function from the Siamese primes to the Siamese primes is either a homeomorphism or hypercontinuous on the whole domain under the induced topology as a subspace of the metric space R!

Psst..we are speaking Fields medal here, aren't we..?

 

[Excalibur1152]

That does not follow. Yes "the x-value can never equal the x-coordinate of the limit (c)". That comes from the "0<" in $0< |x- c|< \delta$ part of the definition of limit.

But it does NOT follow from that "there will always be a distance between y and L". For example, if y= f(x) is a constant, f(x)= L for all x, then y= L no matter how far x is from c.

True, didn't think of that.

 

[LCKurtz]

There is so much beauty in the set of Siamese primes the world just is longing to be exposed to..

Here's a new conjecture: If p and q are distinct siamese primes, then pⁿ+qⁿ is an ordinary prime for positive integers n.

I have managed to prove it for n = 1 and 2

But I'm having a lot of trouble with the induction step. This prime number stuff is soooo difficult.

 

[arildno]

Here's a new conjecture: If p and q are distinct siamese primes, then pⁿ+qⁿ is an ordinary prime for positive integers n.

I have managed to prove it for n = 1 and 2

But I'm having a lot of trouble with the induction step. This prime number stuff is soooo difficult.

Hmm..I'm not sure that will work in its generality, Kurtz.

However, there are far more beautiful results that actually ARE proveable:
Given p,q distinct siamese primes, not only is $$p^{p}+q^{q}$$ prime, but ALSO $$p^{q}+q^{p}$$!
This kind of deep symmetry of the siamese primes more than makes up for that their set isn't closed under multiplication. (But again, that property is rather rare among primes in general, so we shouldn't expect it..)

 

[disregardthat]

Psst..we are speaking Fields medal here, aren't we..?

Good idea, if we could only turn the noses of those crackpot mathematicians towards this field of study somehow...

 

[hemiboso]

Sequencing of the Siamese Primes is deconstructed here using modulo 30 analytics: http://www.primesdemystified.com/siameseprimes.html