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Finding the limit of a quotient as x goes to minus infinity

  1. Sep 27, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the limit
    $$\lim_{x\to-\infty} \frac{\sqrt{9x^6 - x}}{x^3 + 9}$$

    2. Relevant equations
    N/A

    3. The attempt at a solution
    To solve this, I start off by dividing everything by ##x^3##:
    Numerator becomes ##\frac{\sqrt{9x^6 - x}}{x^3} = \sqrt{\frac{9x^6 - x}{x^6}} = \sqrt{9 - \frac{1}{x^5}}##
    Denominator becomes ##\frac{x^3 + 9}{x^3} = 1 + \frac{9}{x^3}##
    As ##x## approaches ##-\infty##:
    Numerator becomes ##\sqrt{9 - 0} = \sqrt{9} = 3##
    Denominator becomes ##1 + 0 = 1##
    So, the entire limit should evaluate to ##\frac{\sqrt{9}}{1} = 3##. Yet this is not the case. What am I doing wrong?
     
  2. jcsd
  3. Sep 27, 2016 #2

    andrewkirk

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    That is not correct if ##x## is negative. How should you adjust it for ##x## negative?
     
  4. Sep 27, 2016 #3

    epenguin

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    Yet this is not the case. Why not? That's the only thing wrong I can see.
     
  5. Sep 27, 2016 #4
    Ah, now I remember. Because ##x## is approaching minus infinity, it will be negative: ##x^3## should be negative if ##x## is negative, but ##x^6## would return a positive number. So, I just need to switch the sign in that equation, right? ##\frac{\sqrt{9x^6 - x}}{x^3} = -\sqrt{\frac{9x^6 - x}{x^6}}## Therefore the answer is -3, not 3.

    I don't know why, but I always forget to consider positives/negatives when working around square roots. Thanks a bunch for your clarification!
     
  6. Sep 27, 2016 #5

    epenguin

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    Hope you don't feel so alone now. :redface:
     
  7. Sep 28, 2016 #6

    Mark44

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    ##\frac{\sqrt{9x^6 - x}}{x^3} = \frac{|x^3|\sqrt{9 - \frac 1 {x^5}}}{x^3}##
    The fraction ##\frac{|x^3|}{x^3}## evaluates to -1 if x < 0, and 1 if x > 0.
     
  8. Sep 28, 2016 #7

    Ray Vickson

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    Sometimes it helps to convert everything to positives, by putting ##x = -t## and taking ##t \to +\infty##. That does not really change anything, but it eliminates one possible source of confusion/error.
     
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