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Homework Help: Oblique asymptote question

  1. Apr 25, 2008 #1
    I am trying to draw the graph of

    y = x + \sqrt{|x|}

    Can I say that as x approaches infinity, y approaches x? That would mean that the function has an oblique asymptote at the line y=x but I'm not sure.

    Thanks for any help!
  2. jcsd
  3. Apr 25, 2008 #2


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    y=x is an asymptote if (and only if):
    y-x approaches zero as x approaches infinity.

    Try subtracting x from both sides of your equation, and see whether the right-hand-side expression approaches zero or not.

    Alternatively, you could plug in successively larger values of x into your expression (i.e, 100, then 1000, then 10,000).
    Does the difference between the value and the value of x get larger or smaller?

    Eg., for x = 100:
    100 + sqrt(100) = ?
    This number is ____ larger than 100.
    Repeat for 1000, then 10,000.
  4. Apr 25, 2008 #3
    Thanks for the reply. Thats what I was thinking, the difference gets larger as x gets larger. But then I thought, when you have a function like x^2 + x, it behaves more and more like x^2 as x gets larger. This function behaves more like x as x gets larger, doesn't it?
  5. Apr 25, 2008 #4


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    Both statements are true, these functions "behave more like" x^2 and x, respectively. However, to be an asymptote is a more stringent requirement.

    It's likely that, if encountered in a physics or engineering application, you'd be completely justified in approximating the function simply by y=x for large x.
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