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Math Help - Finding the domain and asymototes of a function

  1. #1
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    Finding the domain and asymototes of a function

    Hi..I'm not completely clueless on this problem I would just like to see if my answers are correct. The problem is: let f be the function given by f(x)=x/sqrt((x^2)-4)
    A) find the domain of f
    B) find the vertical asymototes of f
    C) find the horizontal asymototes of f
    When I attempted this problem algebraic ally (which is what the directions said) I got the answer of (-infinity,-2),(2,infinity) for the domain, vertical asymototes at x=2 and x=-2 and the horizontal asymptote to be y=o because the degree of the denominator is bigger than the numerator...however thinking twice about those rules I now think that only applies to rationalized functions..which this is not...annd when I graphed it on my calculator I got a corner hyperbola with what appeared to be a vertical asymptote at x=I and horizontal asymototes at y=1 and y= -1....please help!! Thanks!-Kathryn
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  2. #2
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    Re: Finding the domain and asymototes of a function

    Your thinking is on the right track but there is just one small mistake:

    because the degree of the denominator is bigger than the numerator
    The degrees are in fact equal, because of the square root. In the denominator, as x approaches infinity or negative infinity, the -4 term matters less and less and eventually not at all, as it gets swamped by the increasingly large x^2 term, which is square rooted. So essentially, as x approaches positive infinity, the function behaves like \frac{x}{\sqrt{x^2}} = \frac{x}{x} = 1. At negative infinity, this is similar, except the numerator will be negative while the denominator will always be positive, so it approaches -1. This accounts for the horizontal asymptotes.

    And there are actually vertical asymptotes at x = \pm2, you were correct there.
    Last edited by SworD; September 12th 2012 at 08:43 PM.
    Thanks from Kathrynm77
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    MHF Contributor MarkFL's Avatar
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    Re: Finding the domain and asymototes of a function

    You did fine for the domain and the vertical asymptotes, for the horizontal asymptotes, consider:

    \lim_{x\to-\infty}f(x)

    \lim_{x\to\infty}f(x)

    What do you find when you evaluate these limits?
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  4. #4
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Finding the domain and asymototes of a function

    \lim_{x\to \infty } \, f(x) \to +1

    \lim_{x\to -\infty } \, f(x) \to -1
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    Re: Finding the domain and asymototes of a function

    Oh ok..it makes sense now to use limits..thanks so much!!
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