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The meaning/definition of Limit

  1. Apr 14, 2009 #1
    If for every number E>0, there exists a corresponding number D>0 such that for all x
    |x-x0|<D >> |f(x)-L|<E
    Then L is a limit

    what is precisely mean of "corresponding number"?
    and how can that "correspondance" assure me that the limit exists?

    how can a number be corresponding to another number?
    I know how can a number be equal/less than/greater than a number, but how can it be corresponding to another number?

    I think I understand the definition now, except this "essential?" part
  2. jcsd
  3. Apr 14, 2009 #2
    It just means that if someone tell you E>0, you can tell them D>0 so that the second part is true.
  4. Apr 14, 2009 #3
    Your intention is to foil the following series of arguments by coming up with deltas that work for every scenario:
    "You say that f(x) approaches the value L near x=a, but I bet you can't give me an f(x) that is 0.000001 away from L!"
    "Of course I can. Just take any x in the interval (a - D, a + D)!"
    "Oh okay. but I bet you can't give me an f(x) within 10-10000 of L!"
    "No problem. Let x be in the interval (a - D1, a + D1)!"
    "Hmm. What about ..."
    and so on. The challenges are the epsilons and the responses are the deltas. You can usually formulate an epsilon-delta argument so that the deltas are some function of the epsilon challenges, or argue by inequalities.
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