# Formal limit definition

1. Oct 9, 2004

### mfk_1868

i didnt understand delta definiton anyone can explain?

2. Oct 9, 2004

### arildno

Post it, and pinpoint what you don't understand.

3. Oct 9, 2004

### mfk_1868

i mean i didnt know it exactly but only thing i know it uses triangle inequality

4. Oct 9, 2004

### arildno

You have a book, right?
In there is a definition of the limit.
Post that, and say what you didn't understand.

5. Oct 9, 2004

### mfk_1868

i have a book and at there there is epsilon definition. but teacher told a different method named delta using triangle inequality and i am asking this.i dont have any other info.

6. Oct 9, 2004

### cogito²

We say that a sequence of real numbers $$(s_n)$$ converges to $$s$$ (a real number) if given an $$\epsilon > 0$$ there exists a natural number $$N$$ such that $$n \geq N$$ implies that $$|s_n - s| < \epsilon$$. What don't you understand about this definition?

7. Oct 10, 2004

### mfk_1868

where did you used the triangle inequality i know the epsilon definiton and understood it but there is a delta definition.

8. Oct 10, 2004

### cogito²

Do you mean the limit of a function? If $$f$$ is a real-valued function we say that $$\lim_{x \to a} f(x) = L$$, if given $$\epsilon > 0$$ there exists a $$\delta > 0$$ such that $$|x - a| < \delta$$ imply that $$|f(x) - L| < \epsilon$$.

Is that the definition you don't understand? Are talking about metric spaces?

???

9. Oct 10, 2004

### Hurkyl

Staff Emeritus
I've never heard of an epsilon definition of a limit, nor have I heard of a delta definition of a limit. I have, however, heard of the epsilon-delta definition of a limit, which is the standard definition. This definition does not make use of the triangle equality.

10. Oct 10, 2004

### mfk_1868

yes can you show me a proving example. for example lim x->3 x^2/5 how to proof this.

11. Oct 10, 2004

### arildno

What do you mean by "proving example"??
This is from the DEFINITION of a limit ; not some alleged proposition of properties limits may have!
Do you understand that difference?

12. Oct 10, 2004

### mfk_1868

for example there is a question that it gives you a limit equation then says prove this using delta method. I meaned this.

13. Oct 10, 2004

### arildno

What Is A Definition?

14. Oct 11, 2004

### matt grime

So, you've some function and want to show f(x) tends to some given value L, say, as x tends to w? (using the full epsilon delta argument)

well, sorry, we can't do that without knowing what the question is. there is no method that works always, it depends on the question, though they often have the same underlying idea.

15. Oct 11, 2004

### mfk_1868

it was a quiz question and teacher posted it on internet link is here http://www.fen.bilkent.edu.tr/~otekman/calc/quiz101.pdf [Broken] first question number is 6 at the top of page.(look at the solution he uses triangle inequality)

Last edited by a moderator: May 1, 2017
16. Oct 12, 2004

### matt grime

he's perfectly entitled to use the triangle inequality. it's very useful.

i've not actually looked at the pdf but i can give you a proof of a property limits using the triangle inequality.

Suppose a(n) is a sequence tending to a, and b(n) tends to b, then a(n)+b(n) tends to a+b

proof:

consider

|a(n)+b(n)-a-b| = | a(n)-a + b(n)-b| <= |a(n)-a| + |b(n)-b| ***

by the triangle inequality.

Given e (epsilon) let N be chosen such that |a(n)-a| < e/2 and |b(n)-b| <e/2 for all n>N, which we may do since a(n) tends to a and b(n) tends to b.

then *** is less than or equal to e/2+e/2 =e, as we were required to show.

is that what you were thinking of?

17. Oct 12, 2004

### HallsofIvy

But look at problem 6 closely. It doesn't deal with "limits" at all.

The professor has defined a thing he calls "mimits" by reversing the roles of ε and δ

"The real number M is "mimit" of f at x0 if for every δ>0 there exist ε> 0 such that if 0<|x- x0|< δ then |f(x)- M|< ε"

Since in this case, you are given[\b] δ and asked to show that ε exists, finding ε is really just a matter of replacing x with x0+/- δ and calculating ε.

MFK 1868: We were having trouble understanding this because this is NOT a standard notation. Your professor was giving you a problem that requires you to look at the ideas from another point of view.