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Simple limits problem

  1. Jul 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Find
    [tex]\lim_{x\rightarrow \infty} (\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})[/tex]


    2. Relevant equations



    3. The attempt at a solution
    Rewriting the given expression,
    [tex]\sqrt{x}\left(\sqrt{1+\sqrt{\frac{1}{x}\left(1+\frac{1}{\sqrt{x}}\right)}}-1\right)[/tex]
    What should I do with the sqrt(x) outside? :confused:

    Any help is appreciated. Thanks!
     
  2. jcsd
  3. Jul 9, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    A common way to approach limits of the type of ##\sqrt{x+f(x)}-\sqrt{x}## is a multiplication with ##\displaystyle 1=\frac{\sqrt{x+f(x)}+\sqrt{x}}{\sqrt{x+f(x)}+\sqrt{x}}##. This does not change the limit (as you multiply with 1), but you can simplify the numerator a lot.
     
  4. Jul 9, 2013 #3

    Curious3141

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    Homework Helper

    I wanted to post earlier but kept messing up my algebra.

    Call the expression ##y##. Find ##y.(\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt x)##.

    mfb has suggested pretty much the same thing.
     
  5. Jul 9, 2013 #4
    Thanks mfb! :smile:
     
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