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Homework Help: Limit to infinity

  1. Oct 29, 2012 #1
    1. The problem statement, all variables and given/known data
    [tex]\lim_{x \to \infty} \frac{\sqrt{9x^6 - x}}{x^3 + 1}[/tex]

    2. Relevant equations

    3. The attempt at a solution
    \frac{\sqrt{9x^6 - x}}{x^3 + 1} \cdot \frac{\frac{1}{x^3}}{\frac{1}{x^3}} = \\ \frac{\frac{\sqrt{9x^6 - x}}{x^3}}{1 + \frac{1}{x^3}} =\\ \frac{(1 + \frac{1}{x^3})(\sqrt{9x^6 - x})}{x^3} = \\ \frac{(1 + \frac{1}{x^3})(\sqrt{9x^6 - x})}{x^3} \cdot \frac{\frac{1}{x^3}}{\frac{1}{x^3}} = \\ \frac{(1 + \frac{1}{x^3})(\sqrt{9x^6 - x})}{x^3}[/tex]

    And it just repeats over and over again and I can't find anything to divide by without destroying the work I've already done. What am I supposed to do in a loop and there's nothing to divide by?
  2. jcsd
  3. Oct 29, 2012 #2
    You have to keep better track of the order of calculations.. Remember that a/(b/c) ≠ (a/b)/c.
  4. Oct 29, 2012 #3


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    Staff Emeritus
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    Gold Member

    The next line is not equivalent to the above.

    Then use the fact that [itex]x^3=\sqrt{x^6}\ .[/itex]
  5. Oct 29, 2012 #4


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    Staff: Mentor

    Here is a hint: Try to combine the 1/x^3 with your square root.
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