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Limit of trigometric function with x-sqrt/x-sqrt

  1. Oct 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the value of lim x→∞

    [x-sqrt(x^2+5x+2)] / [x-sqrt(x^2+(x/2)+1)]

    Answer is 10.

    2. Relevant equations


    3. The attempt at a solution

    I tried to multiply by the conjugate and got

    [5x+2] / [x-sqrt(x^2+5x+2)] * [x+sqrt(x^2+(x/2)+1)]

    but then I'm still stuck because I still get ∞ as the answer.

    I also tried to divide both the top and bottom by x. Then I get [1-sqrt(1+(5/x)-(2/x^2))]/[1-sqrt(1+(x^3/2)+(1/x^2)) = 0/0 which is incorrect
     
  2. jcsd
  3. Oct 21, 2012 #2

    SammyS

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    Hello mesasi. Welcome to PF !

    Also multiply the numerator & denominator by the conjugate of [x-sqrt(x^2+(x/2)+1)]
     
  4. Oct 21, 2012 #3
    After I do and divide by x on both sides I get





    [itex]\frac{(-5x-2)(1+\sqrt{1+(1/2x)+(1/x^2)}}{((-x/2)-1)(1+\sqrt{1+(5/x)+(2/x^2)}}[/itex]


    then I get [itex]\frac{-5x}{(-x/2)}[/itex]+2

    which simplifies to 12 which is still not 10? What am I doing wrong?
     
  5. Oct 21, 2012 #4

    SammyS

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    [itex]\displaystyle \frac{-5x-2}{1+(1/2x}\ne\frac{-5x}{(-x/2)}+2[/itex]
     
  6. Oct 21, 2012 #5
    (5x+2)/((x/2)+1) * (2/2) = 10

    Thank you!
     
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