Math Help - Limit question,

1. Limit question,

Hey guys not sure where to go with this, I am generally ok with delta epsilon questions but Im stuck here. I have a Summer test Thursday and supposedly something similar will come up.

Suppose that f
(x) approaches L as x approaches a.
Show that there corresponds to each positive number E (epsilson) a positive number d (delta) such that,
|f(x1)-f(x2)| < E

for every choice of x1and x2 in
I( a, d ).

2. Re: Limit question,

Originally Posted by MathJack
Hey guys not sure where to go with this, I am generally ok with delta epsilon questions but Im stuck here. I have a Summer test Thursday and supposedly something similar will come up.
Suppose that f(x) approaches L as x approaches a. Show that there corresponds to each positive number E (epsilson) a positive number d (delta) such that,
|f(x1)-f(x2)| < E
for every choice of x1and x2 in I( a, d ).
There is a real puzzle contained in your question.
That question asks you to prove a what the vast majority of Authors and Textbooks use as the definition of limit.

So your text material must use a different definition. What definition of limit do you have?

3. Re: Limit question,

Sorry its on a pdf,

'We say that f has a limit L at a if, in the case of each positive number E, a positive number d can be found such that |f(x) - L| < E for every x element I'(a,d)

If f has a limit L at a we write

lim f(x) = L or f(x) approaches L as x approaches a
'

And thats it

4. Re: Limit question,

A better layout of the definition and question are on the sheet. Sorry about the delay, wasnt sure if it would come up

5. Re: Limit question,

Originally Posted by MathJack
Sorry its on a pdf,
'We say that f has a limit L at a if, in the case of each positive number E, a positive number d can be found such that |f(x) - L| < E for every x element I'(a,d)
If f has a limit L at a we write
lim f(x) = L or f(x) approaches L as x approaches a

Sorry, I will not open an attached file.
If you are to lazy to type it out using LaTeX, then you may be out of luck.

6. Re: Limit question,

I do not know how to use LaTex yet

'We say that f has a limit L at a if, in the case of each positive number E, a positive number d can be found such that |f(x) - L| < E for every x element I'(a,d)

If f has a limit L at a we write

lim f(x) = L or f(x) approaches L as x approaches a

7. Re: Limit question,

Originally Posted by MathJack
'We say that f has a limit L at a if, in the case of each positive number E, a positive number d can be found such that |f(x) - L| < E for every x element I'(a,d)
If f has a limit L at a we write lim f(x) = L or f(x) approaches L as x approaches a
That is in a deleted neighborhood of $a$.
That simply means that if $0<|x-a|<\delta$ then $|f(x)-L|<\epsilon .$

What happen if we drop the deleted neighborhood part?

Suppose it is true on the non-deleted neighborhood, is it true on the deleted neighborhood ?

8. Re: Limit question,

Hi,
Here's a proof of what you want. The triangle inequality is used; you'll definitely see this usage again as you progress. However, in asking questions, you should be very careful to ask the exact question. Your original post left out the deleted neighborhood which makes the statement false.

9. Re: Limit question,

If we drop the deleted neighborhood part then |f(x)-L| < E is always true, no restrictions?

So if it is true on the non-deleted neighborhood then for the same values it must be true on the deleted neighborhood. (this can be said for an infinite number of larger NDNs). And if true on the DN then |f(x1) - f(x2)| < E must be true?

10. Re: Limit question,

Originally Posted by johng
Hi,
Here's a proof of what you want. The triangle inequality is used; you'll definitely see this usage again as you progress. However, in asking questions, you should be very careful to ask the exact question. Your original post left out the deleted neighborhood which makes the statement false.

Cheers johng, triangle inequality..bingo