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Homework Help: Not so Basic Limit!

  1. Apr 11, 2010 #1
    1. The problem statement, all variables and given/known data

    Find this crazy limit using algebraic manipulations. I've tried quite a bit of stuff and keep getting lost, can you recommend anything?

    The answer is 2.

    2. Relevant equations

    [tex] \lim_{x \to - 1} \frac{108(x^2 \ + \ 2x)(x \ + \ 1)^3}{(x^3 \ + \ 1)^3(x \ - \ 1)} [/tex]


    3. The attempt at a solution

    [itex] \lim_{x \to - 1} \frac{108(x^2 \ + \ 2x)(x \ + \ 1)^2(x \ + \ 1)}{(x^3 \ + \ 1)^3(x \ - \ 1)} [/itex]

    [itex] \lim_{x \to - 1} \frac{108(x^2 \ + \ 2x)(x ^2\ + \ 2x \ + \ 1)(x \ + \ 1)}{(x^3 \ + \ 1)^3(x \ - \ 1)} [/itex]

    [itex] \lim_{x \to - 1} \frac{108(x^2 \ + \ 2x)(x ^3\ + \ 3x^2 \ + \ 3x \ + \ 1)}{(x^3 \ + \ 1)^3(x \ - \ 1)} [/itex]

    If I were to keep going expanding I'll get x^10 on the bottom and x^5 on the top, it just gets so hairy & I can't find any commonality.
     
  2. jcsd
  3. Apr 11, 2010 #2
    Simplify it to

    [tex] \frac{108 x (2+x)}{(x-1) \left(1-x+x^2\right)^3} [/tex]
     
  4. Apr 11, 2010 #3
    Ahh!!! I should have just realised to factor x³ + 1!!! So simple now thank you :)
     
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