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Math Help - Limits at infinity

  1. #1
    Junior Member ImaCowOK's Avatar
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    Limits at infinity

    Finding the limit of (9x + 5)/(7x + 9) as x approaches ∞.

    So, I have to divide the numerator and denominator by the highest power of x in the denominator? If so, that would be 1. Therefore, I would have to divide both the numerator and denominator by x/1?

    Is that correct?

    Then I would get (9 + 5/x)/(7 + 9/x). Where would I go from there?

    Okay, I got it. 5/x and 9/x become zero, so the answer is 9/7.

    Do you always divide by the highest power of x in the denominator, and never the numerator?
    Last edited by ImaCowOK; August 29th 2010 at 10:44 AM.
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  2. #2
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    Quote Originally Posted by ImaCowOK View Post
    Finding the limit of (9x + 5)/(7x + 9) as x approaches ∞.
    Do you always divide by the highest power of x in the denominator, and never the numerator?
    None of us likes to answer a question that contains always
    These are rules of thumb and nothing more.
    If you have the ratio of two polynomials that is a correct rule of thumb.
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  3. #3
    Junior Member ImaCowOK's Avatar
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    I know we are not supposed to generalize in math, I've heard it said many times already. I would like to know the rules, if when solving these I will have to use the highest power of x in the denominator. If I have one with x in the numerator, but the highest power in the denominator is just x - do I divide by x or x?
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  4. #4
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    The point is: there are no hard and fast rules.
    If we have two polynomials, say \dfrac{x^2+x+1}{x+1} then divide by x.
    The limit is infinity.
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  5. #5
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    for functions formed by the ratio of polynomials, there are three cases ...

    degree of numerator > degree of denominator ... as x \to \infty the value of the rational function increases w/o bound.

    degree of numerator < degree of denominator ... as x \to \infty the value of the rational function approaches 0.

    degree of numerator = degree of denominator ... as x \to \infty the value of the rational function approaches the ratio of the leading coefficients.
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  6. #6
    Junior Member ImaCowOK's Avatar
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    Quote Originally Posted by Plato View Post
    there are no hard and fast rules.
    I know that.

    You resolved that last question: I would divide by x in that case.
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  7. #7
    Junior Member ImaCowOK's Avatar
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    Thanks everyone. Add anymore info as needed.
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