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Math Help - Finding horizontal asymptote

  1. #1
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    Finding horizontal asymptote

    Equation: x^2 - 64 / ((3x+4)(x-5))

    The horizontal asymptote is 1/3. I graphed this and for very large values of x, the graph equals appromixmately .354 and a little up. How is 1/3 an asymptote then? My books ays to make x equal a very large positive number, and then it will be (x^2)/(3x^(2)), which is 1/3. Am I looking at the graph wrongly?
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  2. #2
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    Re: Finding horizontal asymptote

    Quote Originally Posted by benny92000 View Post
    Equation: x^2 - 64 / ((3x+4)(x-5))

    The horizontal asymptote is 1/3. I graphed this and for very large values of x, the graph equals appromixmately .354 and a little up. How is 1/3 an asymptote then? My books ays to make x equal a very large positive number, and then it will be (x^2)/(3x^(2)), which is 1/3. Am I looking at the graph wrongly?
    \frac{x^2 - 64}{3x^2-11x-20} = \frac{1 - \frac{64}{x^2}}{3 - \frac{11}{x} - \frac{20}{x^2}}

    what happens to each term in the rational expression as x \to \infty ? what is left unchanged?
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  3. #3
    Super Member TheChaz's Avatar
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    Re: Finding horizontal asymptote

    Quote Originally Posted by skeeter View Post
    \frac{x^2 - 64}{3x^2-11x-20} ...}
    If I may borrow your markup...

    Another way to think about this is to think of how "powerful" each term in a polynomial is. In the numerator, x^2 is more "powerful" than -64, since x^2 increases more rapidly (... well, -64 doesn't increase or decrease). So when x is large*, the x^2 term will dominate.

    Likewise for the denominator. For x "sufficiently large", the 3x^2 term will dominate, so your fraction will be - roughly:

    \frac{x^2}{3x^2}, and you can see that this is 3.
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  4. #4
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    Re: Finding horizontal asymptote

    That's how I would solve a limit. How do I know to make x approach positive infinity and say not negative infinity?
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  5. #5
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    Re: Finding horizontal asymptote

    Any terms involving x to an odd power go to 0 so it doen't matter if x goes to infinity or negative infinity.
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