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Homework Help: Limit question with Substituting

  1. Oct 28, 2011 #1
    Fine the limit of (x^1000 - 1)/(x - 1) as x approaches 1.

    Solution:


    (x^1000 - 1)/(x - 1) = (x^999 + x^998 + ... + x + 1)(x - 1)/(x - 1)

    = (x^999 + x^998 + ... + x + 1)

    Substituting x = 1 I get....

    Line (1):1 + 11 + 12 + ..... + 1999 = 1 + 999 = 1000

    Is there any way to prove Line (1): I know it is obvious but I want to prove it mathematically and with out obviously counting it on my fingers.
     
  2. jcsd
  3. Oct 28, 2011 #2
    I guess I could say 1 + 1(999) = 1000
     
  4. Oct 28, 2011 #3
    If any one has a better way of solving it please let me know, I want to learn all I can.
     
  5. Oct 28, 2011 #4

    SammyS

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    You could use synthetic division.
     
  6. Oct 28, 2011 #5

    Simon Bridge

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    It's one of the surprising solutions that the limit as x --> 1 of (x^N -1)/(x-1) = N ... in the limit, the numerator is N times bigger than the denominator.

    The question is to prove the expansion - you can demonstrate it simply enough by multiplying out the brackets so it is not clear what you mean when you want a non finger-counting method.
     
  7. Oct 28, 2011 #6

    Dick

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    You could use l'Hopital's theorem, if you have that.
     
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