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Thread: Limits... again =)

  1. #1
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    Limits... again =)

    This time i have a question that says, show that lim(x->a) f(x) exists and find it. where i am given lim(x->a) ((x^n)f(x)-a)/(x-a) = 1. Dont bother going through how to find it, I can do that all i need is some clues or hints as to how i am going to SHOW it exists before finding it.
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  2. #2
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    Is that the entire question? Any restrictions on n or a?

    You can approach it this way. Let:
    g(x) = \frac{x^{n}f(x) - a}{x-a} (Note that: \lim_{x \to a} g(x) = 1

    Solve for f(x):
    g(x)(x-a) = x^{n}f(x) - a
    f(x) = \frac{g(x)(x-a)+a}{x^{n}}

    \lim_{x \to a} f(x) = \lim_{x \to a} \frac{g(x)(x-a)+a}{x^{n}}

    If you can show that the right hand limit exists then you're good but of course there's the issue of what happens if a = 0 or a = 0 AND n = 0.
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