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Do points have limits?

  1. May 22, 2015 #1
    Can someone rephrase the title question into something more meaningful in terms of Calculus/Analysis?
     
  2. jcsd
  3. May 22, 2015 #2
    Possibly kindly answer it as well. Thanks.
    - blue
     
  4. May 22, 2015 #3

    FactChecker

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    A single point does not but a series of points can approach a limit.
     
  5. May 23, 2015 #4

    Fredrik

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    Suppose that ##x_1,x_2,\dots## is a sequence of points. A point ##x## is said to be a limit of that sequence if every open neighborhood of ##x## (i.e. every open set that contains ##x##) contains all but a finite number of the points in the sequence.
     
  6. May 23, 2015 #5

    pwsnafu

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    There is a difference between the individual point ##x## and the sequence ##(x,x,x,\ldots)##. The former the does not have a limit but the latter does (and the limit is ##x##).
     
  7. May 24, 2015 #6

    HallsofIvy

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    Just to clarify, here we would say that x is the limit, not that x "has" a limit!
     
  8. Jun 7, 2015 #7

    disregardthat

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    Note that sequences may have many limits. If your topological space is not hausdorff, this may happen.

    This is slightly imprecise depending on interpretation. It should either say all but a finite number of terms in the sequence, since the sequence might eventually stabilize.

    Not sure how to interpret this, but I'd point out that the limit points of the one-point set ##\{x\}## are exactly the limit points of the sequence (x,x,...)
     
  9. Jun 12, 2015 #8

    Stephen Tashi

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    The terms "limit of a sequence of points" and "limit point of set of points" have different meanings. A set of points (not necessarily arranged as a sequence) can have many "limit points". https://en.wikipedia.org/wiki/Limit_point
     
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