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Homework Help: Limits to infinity

  1. Oct 27, 2012 #1
    1. The problem statement, all variables and given/known data
    [tex]\lim_{x \to \infty} \frac{2x}{\sqrt{x+2} + \sqrt{x}}\\\\\\[/tex]

    2. Relevant equations

    3. The attempt at a solution
    [tex]\lim_{x \to \infty} \frac{2x}{\sqrt{x+2} + \sqrt{x}}\\\\\\ \lim_{x \to \infty} \frac{\frac{2x}{x}}{\sqrt{\frac{x}{x}+\frac{2}{x}} + \sqrt{\frac{x}{x}}}\\\\\\ \lim_{x \to \infty} \frac{2}{\sqrt{1 + \frac{2}{x}} + \sqrt{1}}\\\\\\ \lim_{x \to \infty} \frac{2}{\sqrt{1} + \sqrt{1}}\\\\\\ \lim_{x \to \infty} \frac{2}{1 + 1} = \frac{2}{2} = 1\\\\\\[/tex]

    But this is incorrect. Where have I done my work incorrectly?
  2. jcsd
  3. Oct 27, 2012 #2


    User Avatar
    Homework Helper

    The denominator in the second step is wrong.

    [tex]\frac{\sqrt{f(x)}}{x} = \sqrt{\frac{f(x)}{x^2}}[/tex]

    (for positive x, of course).
  4. Oct 27, 2012 #3
    I don't understand why you have to divide by x^2 and not just x.
  5. Oct 27, 2012 #4


    User Avatar
    Science Advisor

    Its because SQRT(x^2) = x if x >= 0.
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