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Calculate the limit help

  1. Mar 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Calculate the limit of
    [tex]
    \lim_{x\rightarrow \infty} \frac{\sqrt{x^{2}+5} - x}{\sqrt{x^{2}+2} - x}
    [/tex]

    2. Relevant equations
    -

    3. The attempt at a solution
    Neither multiplying with the conjugate nor trying to break out x helps me, as I'm left with "0/0" in those cases.
     
  2. jcsd
  3. Mar 10, 2009 #2

    lanedance

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    Re: $\lim_{x\rightarrow\infty} \frac{\sqrt{x^{2}+5} - x}{\sqrt{x^{2}+2} - x}$

    do you know l'hopitals rule? could help here i think
     
  4. Mar 10, 2009 #3
    Re: $\lim_{x\rightarrow\infty} \frac{\sqrt{x^{2}+5} - x}{\sqrt{x^{2}+2} - x}$

    While I do know l'Hopitals rule, we have not yet covered it in the course. The problem should be solved without using (sadly).
     
  5. Mar 10, 2009 #4

    lanedance

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    Re: $\lim_{x\rightarrow\infty} \frac{\sqrt{x^{2}+5} - x}{\sqrt{x^{2}+2} - x}$

    how about this, multiply by both conjugates of the numerator & denominator to get:

    [tex]
    \lim_{x\rightarrow \infty} \frac{5 \sqrt{x^{2}+2} + x}{2 \sqrt{x^{2}+5} +x}
    [/tex]

    already looking in better shape, as its not a difference term that is leading to the zero, which ws the tricky bit, so from here I'd try multiplying through by:

    [tex]
    \lim_{x\rightarrow \infty} \frac{\frac{1}{x}}{\frac{1}{x}}
    [/tex]

    this should change the terms containing x in the numerator & denominator from tending to infinity, to ones tending to zero...
     
  6. Mar 10, 2009 #5
    Re: $\lim_{x\rightarrow\infty} \frac{\sqrt{x^{2}+5} - x}{\sqrt{x^{2}+2} - x}$

    [tex]\lim_{x\rightarrow \infty} \frac{5\sqrt{x^{2}+2} + x}{2\sqrt{x^{2}+5} +x} = \lim_{x\rightarrow\infty} = \frac{5}{2}\frac{x(\sqrt{1+\frac{2}{x}})+1}{x(\sqrt{1+\frac{5}{x}})+1} = \frac{5}{2}\cdot\frac{2}{2} = \frac{5}{2}

    [/tex]

    Cheers!
     
  7. Feb 9, 2012 #6
    Re: $\lim_{x\rightarrow\infty} \frac{\sqrt{x^{2}+5} - x}{\sqrt{x^{2}+2} - x}$

    There were some significant errors in this (highlighted in the

    quote box) post from a user, that I felt one of the corrected

    versions should be shown.
     
  8. Feb 9, 2012 #7

    micromass

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    Re: [itex]\lim_{x\rightarrow\infty} \frac{\sqrt{x^{2}+5} - x}{\sqrt{x^{2}+2} - x}[/it

    This thread is 2 years old. You've been here long enough to know not to necropost.
     
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