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Sketch The Curve

  1. Nov 19, 2005 #1
    Hi guys,

    I know this is a dumb question, but I have to ask. We are supposed to sketch the curve of y=x/(x-1)^2 using the guidelines for curve sketching such as domain, intervals of increase/decrease, concavity....ect. For the most part on this question I think I have done this all right, it really isn't all that hard. But for some reason my asymptotes are a bit messed up.

    I know there is a vertical asymptote occuring at x=1, because having an x value of one makes the denominator undefined. Then to find the horizontal asymptoes I expanded the denominator to get
    y = x/(x^2 -2x+1), then I found the horizontal denominator (limit as it approaches infinity) by taking an x^2 out of the whole thing, leaving me with a horizontal asymptote at y=0, which for the right half of the curve is true. But the left half of the curve does cross the axis at (0,0). How do I account for the fact that there is not a horizontal tangent at y=0 on the left side of the graph? I hope I explained my situation clearly. Thanks for any help in advance.
     
  2. jcsd
  3. Nov 19, 2005 #2

    Tide

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    The function approaches the x axis from below as x goes to negative infinity.
     
  4. Nov 19, 2005 #3
    OK, I get the statement you just made. But how would I go about showing that mathematically?
     
  5. Nov 19, 2005 #4

    Tide

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    For x < 0, the function is negative and there are no other zeros of the function.
     
  6. Nov 20, 2005 #5
    Ok so because I have found that there is a horizontal asymptote at x=0 I can say that as x approaches both infinity and negative infinity the graph approaches but never reaches zero, except for the intercept at (0,0) ?
     
  7. Nov 20, 2005 #6

    Tide

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    Well, there's a horizontal asymptote at y = 0 and not at x = 0 and since the function is negative for all negative values of x so the graph approaches the asymptote from below for negative x but from above for positive x.
     
  8. Nov 20, 2005 #7
    Ok I think I got it now, thanks
     
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