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Math Help - Asymptotes of y = x - arctanx

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    Asymptotes of y = x - arctanx

    Sketch the graph of y = x - arctanx, showing its asymptotes.

    Now I got the correct shape of the graph, but my asymptotes were wrong.

    I dont understand how you would get the asymptote of y = x + π/2 and y = x - π/2. Could someone please explain?
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    Remember that tangent is not a one-to-one function (ie, it sends multiple points in its domain to the same point in its range). Thus, in order for arctangent to be a function, its range has to be restricted to  -\frac{\pi}{2} < x < \frac{\pi}{2} , otherwise it would be multiply defined (without the restriction, where, for instance, would arctan(0) go?). So, to find the asymptotes, we need to find where the graph does not pass.

    We know that  |arctan(x)| < \frac{\pi}{2} , so where are those asymptotes?
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    Quote Originally Posted by Math Major View Post
    We know that  |arctan(x)| < \frac{\pi}{2} , so where are those asymptotes?
    I know that arctanx has horizontal asymptotes of y = π/2 and y = -π/2 (inversion of tanx's undefined x values) but I dont see any x values in y = x - arctanx which would make it undefined?
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    If you are talking about vertical asymptotes, it has no vertical asymptotes.
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    Quote Originally Posted by HallsofIvy View Post
    If you are talking about vertical asymptotes, it has no vertical asymptotes.
    Yeah thats what I meant, that it has no vertical/horizontal asymptotes, which is why I wanted to know, how can you tell that y = x - arctanx has oblique asymptotes?
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    Quote Originally Posted by SyNtHeSiS View Post
    Yeah thats what I meant, that it has no vertical/horizontal asymptotes, which is why I wanted to know, how can you tell that y = x - arctanx has oblique asymptotes?
    as x \to \pm \infty , \arctan{x} \to \pm \frac{\pi}{2}.

    so ... as x \to \pm \infty , y = x - \arctan{x} approaches the line y = x
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    Quote Originally Posted by skeeter View Post
    as x \to \pm \infty , \arctan{x} \to \pm \frac{\pi}{2}.

    so ... as x \to \pm \infty , y = x - \arctan{x} approaches the line y = x
    Sorry but I am a bit confused. I understand the first line, but for the second line, how does it approach y=x? I thought the lines it approaches were y= x+/- π/2?
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    Quote Originally Posted by SyNtHeSiS View Post
    Sorry but I am a bit confused. I understand the first line, but for the second line, how does it approach y=x? I thought the lines it approaches were y= x+/- π/2?
    I'll agree with that ...

    as x \to \infty , the function approaches y = x - \frac{\pi}{2}

    as x \to -\infty , the function approaches y = x + \frac{\pi}{2}
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    Quote Originally Posted by skeeter View Post
    I'll agree with that ...

    as x \to \infty , the function approaches y = x - \frac{\pi}{2}

    as x \to -\infty , the function approaches y = x + \frac{\pi}{2}
    One more question, if you use a limit with x tending to +/- infinity, why isnt it y = ∞ - π/2 and y = ∞ + π/2?
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    Quote Originally Posted by SyNtHeSiS View Post
    One more question, if you use a limit with x tending to +/- infinity, why isnt it y = ∞ - π/2 and y = ∞ + π/2?
    there is a difference between the limit of a function and the limit of a function value. \infty \pm \, any \, constant is still \infty
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    Re: Asymptotes of y = x - arctanx

    a
    Last edited by MathCrusader; March 26th 2013 at 08:33 AM.
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