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Precise definition of a limit

  1. Mar 1, 2010 #1
    I would really appreciate if someone could please explain to me how the final result of the proof of a limit actually PROVES the limit? http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preclimsoldirectory/PrecLimSol.html

    I was particularly looking at Solution 2 on this website. I understand how |x-10| was factorized out etc. and therefore it can equal delta... but when it says that if delta = epsilon/3, |x-10| < epsilon/3 and therefore |f(x)-35| is less than epsilon.... how is this true??? I mean we dont even know what epsilon is?

    how does this even prove that the limit is 35
     
    Last edited: Mar 1, 2010
  2. jcsd
  3. Mar 2, 2010 #2
    Does it?

    Do you understand the epsilon-delta argument. It is a two part formal method much used in mathematical analysis. Would this be your difficulty?
     
  4. Mar 2, 2010 #3

    tiny-tim

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    Hi Shaybay92! :smile:

    (have a delta: δ and an epsilon: ε :wink:)
    ε is anything (> 0).

    Continuity is proved if, for any ε, you can find a δ (which depends on that ε) which works. :wink:
     
  5. Mar 2, 2010 #4
    ok but how does this proof tell us that there is a delta for any epsilon? i dont really see how if delta = cepsilon (c is some constant) then that is a proof...
     
  6. Mar 2, 2010 #5

    tiny-tim

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    Hi Shaybay92! :smile:

    (what happened to that δ and ε i gave you? :wink:)
    because, if δ is defined as cε, then for any ε, there is a δ ! :smile:
     
  7. Mar 2, 2010 #6
    so as long as x is less than a particular multiple of ε (cε) then |f(x)-L| < ε?
     
  8. Mar 2, 2010 #7

    tiny-tim

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    That's right. :smile:

    (Warning: although δ is usually a multiple of ε, somtimes it's something more awkward, like ε2 :wink:)
     
  9. Mar 2, 2010 #8
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