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Math Help - [SOLVED] Limit of a composition of functions

  1. #1
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    [SOLVED] Limit of a composition of functions

    I am having trouble verifying the following problem:
    If the functions f and g are defined on (a,\infty) with a\in \mathbb{R}, and where \lim_{x\to \infty} f(x) = L and \lim_{x\to \infty} g(x) = +\infty, prove that \lim_{x\to \infty} (f \circ g)(x) = L

    Here is what I have tried to do
    Suppose (x_{n}) is a sequence defined on (a,\infty) which diverges to \infty.
    At this point I need to show that the sequence (g(x_{n})) diverges to \infty, thereby confirming that the sequence (f(g(x_{n}))) converges to L as n\to \infty.
    How can I show that the sequence (g(x_{n})) diverges?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by dannyboycurtis View Post
    I am having trouble verifying the following problem:
    If the functions f and g are defined on (a,\infty) with a\in \mathbb{R}, and where \lim_{x\to \infty} f(x) = L and \lim_{x\to \infty} g(x) = +\infty, prove that \lim_{x\to \infty} (f \circ g)(x) = L

    Here is what I have tried to do
    Suppose (x_{n}) is a sequence defined on (a,\infty) which diverges to \infty.
    At this point I need to show that the sequence (g(x_{n})) diverges to \infty, thereby confirming that the sequence (f(g(x_{n}))) converges to L as n\to \infty.
    How can I show that the sequence (g(x_{n})) diverges?
    Didn't you post this exact question and get an answer here?
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  3. #3
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    So I did, thank you, I seem to have forgotten. Sorry for the repetition.
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