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Limit question with Substituting

  • Thread starter Miike012
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  • #1
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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.
 

Answers and Replies

  • #2
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I guess I could say 1 + 1(999) = 1000
 
  • #3
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If any one has a better way of solving it please let me know, I want to learn all I can.
 
  • #4
SammyS
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You could use synthetic division.
 
  • #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.
 
  • #6
Dick
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If any one has a better way of solving it please let me know, I want to learn all I can.
You could use l'Hopital's theorem, if you have that.
 

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