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Math Help - Limits

  1. #1
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    Limits

    Suppose \lim_{x\to\infty} f(x)/g(x) = \lim_{x\to\infty} h(x)k(x)/g(x). Can you conclude from this that  \lim_{x\to\infty} k(x) = \lim_{x\to\infty} f(x)/h(x) ?

    I make this step in a somewhat longer proof in my homework, but I am not sure if this is true in general. If it is not true in general, what conditions do I need on the functions?
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  2. #2
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    To me, everything seems consistent with the Properties of Limits, assuming you meant to have parenthesis around all of your functions.
    ie. \lim_{x \to \infty}\left(f(x)/g(x) \right)

    Indeterminate Forms to understand what happens to these limits when dealing with dividing by 0 and infinity and such.
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  3. #3
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    Thanks. The limits are indeed over everything that is to the right of it. I was indeed wondering if everything was correct with 0 and infinity.
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  4. #4
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    consider the following functions:

    f(x) = \frac{1}{x^2}, g(x) = \frac{1}{x}, h(x) = f(x), k(x) = g(x). then

    \lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{h(x)k(x)}{g(x)} = 0, however:

    \lim_{x \to \infty} k(x) = \lim_{x \to \infty} \frac{1}{x} = 0, whereas \lim_{x \to \infty} \frac{f(x)}{h(x)} = 1.

    one cannot apply the multiplicative/quotient properties of limits in this case, unless one is certain that the limits in the denominators are non-zero real numbers. in the example i have given, that is precisely the case for g(x) and h(x).

    for another counter-example, suppose that:

    f(x) = x^2, g(x) = x, h(x) = f(x), k(x) = g(x).

    again, we have:

    \lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{h(x)k(x)}{g(x)} = \infty, but

    \lim_{x \to \infty} k(x) = \lim_{x \to \infty} x = \infty, whereas \lim_{x \to \infty} \frac{f(x)}{h(x)} = 1.

    suppose that: \lim_{x \to \infty} f(x) = L, \lim_{x \to \infty} g(x) = M, \lim_{x \to \infty} h(x) = N, \lim_{x \to \infty} k(x) = P.

    to argue that L/M = NP/M implies L = NP, M cannot be 0, and must be finite.

    and then to argue further that L = NP implies L/P = N, means P cannot be 0, or infinite. you can't divide by 0, or "cancel infinities".
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