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Is there a simpler way to calculate this limit?

  1. Dec 10, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]\lim_{x\to\infty} \frac{(1 - 2x^3)^4}{(x^4 - x^3+1)^3}[/itex]

    2. Relevant equations

    Not really applicable

    3. The attempt at a solution

    After applying L'Hospital's rule 3 times I could just see it getting untidy and I couldn't see my and mistakes in my working so I checked against WolframAlpha which showed that L'Hospital's rule is needed 10 times to get to the solution which seems like a lot of work for a small question. Is there a more efficient method for solving this limit that I'm unaware of?

    EDIT: Sorry if this should've been in the pre calculus section, I wasn't sure.
    Last edited: Dec 10, 2011
  2. jcsd
  3. Dec 10, 2011 #2


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    Staff: Mentor

    Do you know the correct answer? Is it 16?
  4. Dec 10, 2011 #3
    Yes it is
  5. Dec 10, 2011 #4


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    Staff: Mentor

    Take the x3 factor out of the numerator, making it

    (x3)4(1/x3 - 2)4

    Follow the same procedure for the denominator.
  6. Dec 10, 2011 #5
    Thanks so much, never thought of doing that to simplify limits.
  7. Dec 10, 2011 #6


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    Alternatively: Multiply by [itex]\ \frac{\displaystyle\ \frac{1}{x^{12}}\ }{\displaystyle \frac{1}{x^{12}}}\ .[/itex]
  8. Dec 11, 2011 #7
    if you're taking the limit as it goes to infinity you can drop any constants and only keep the highest power of x about since as you go to infinity those are going to be insignificant
    doing this to your equation gives
    [tex]lim \frac{(-2)^4x^{12}}{x^{12}}=lim 16 = 16[/tex]

    lol at wolframalpha applying L'hopitals rule 10 times
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