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Def'n of Limit Point? and limit. 
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#1
Oct2205, 10:49 AM

P: 108

I know that there are different definitions for a limit point .
"A number such that for all , there exists a member of the set different from such that . The topological definition of limit point of is that is a point such that every open set around it contains at least one point of different from ."MATHWORLD Are they all equivalent, when defining "the limit of f"? Or, this may help too, does my definition of a limit sound correct?...(boldfaced variables are vectors) Let f: U>R^n Let a be an element of the reals such that for all delta' >0 there exists an x in U, different than a, such that xa<delta'. We say that lim f(x)=L as x>a, if for every epsilon>0 there exists a delta''>0 so that if xa<delta'' then f(x)L<epsilon. Also, Is it right that I used delta' and delta"? 


#2
Oct2205, 10:53 AM

P: 108

Is it necessary that I write U to be an open set?



#3
Oct2205, 11:11 AM

P: 461

Limit point in this sense is not the same thing at all as the limit of a function as it aproaches a point.
They are two different things. 


#4
Oct2205, 11:20 AM

P: 108

Def'n of Limit Point? and limit.
i know that, but you DO need to define a limit point in order to define the limit. It would really help if you could please look at my definition of a limit.



#5
Oct2205, 11:35 AM

P: 461

To be honest, I'm not sure I understand the question.



#6
Oct2205, 12:48 PM

P: 508

If D is the domain of f, then we see that L is a limit point of f(D). 


#7
Oct2205, 12:59 PM

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P: 39,353

Except for one small point, since x and a are real numbers you mean xa, not xa, Your definition of limit is correct. You really have 2 definitions:
"Let a be an element of the reals such that for all delta' >0 there exists an x in U, different than a, such that xa<delta'." is saying that a is a limit point of the set (of real numbers) U. "We say that lim f(x)=L as x>a, if for every epsilon>0 there exists a delta''>0 so that if xa<delta'' then f(x)L<epsilon." is the definition of limit of the function (which only exists a a limit point of U). No, U does not have to be an open set. Although in that case a would be member of U. If you keep xa then U can be a subset of any R[sup]m[/tex]. In fact, if you use a xa to represent a general metric (distance) function, the definitions are correct for a function between any two metric spaces. If you replace "xa< delta" with "there exist an element of U in every open set containing a". Then your definitions work in any topological space. 


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