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Thread: [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 $\displaystyle f$ and $\displaystyle g$ are defined on $\displaystyle (a,\infty)$ with $\displaystyle a\in \mathbb{R}$, and where $\displaystyle \lim_{x\to \infty} f(x) = L$ and $\displaystyle \lim_{x\to \infty} g(x) = +\infty$, prove that $\displaystyle \lim_{x\to \infty} (f \circ g)(x) = L$

    Here is what I have tried to do
    Suppose $\displaystyle (x_{n})$ is a sequence defined on $\displaystyle (a,\infty)$ which diverges to $\displaystyle \infty$.
    At this point I need to show that the sequence $\displaystyle (g(x_{n}))$ diverges to $\displaystyle \infty$, thereby confirming that the sequence $\displaystyle (f(g(x_{n})))$ converges to L as $\displaystyle n\to \infty$.
    How can I show that the sequence $\displaystyle (g(x_{n}))$ 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 $\displaystyle f$ and $\displaystyle g$ are defined on $\displaystyle (a,\infty)$ with $\displaystyle a\in \mathbb{R}$, and where $\displaystyle \lim_{x\to \infty} f(x) = L$ and $\displaystyle \lim_{x\to \infty} g(x) = +\infty$, prove that $\displaystyle \lim_{x\to \infty} (f \circ g)(x) = L$

    Here is what I have tried to do
    Suppose $\displaystyle (x_{n})$ is a sequence defined on $\displaystyle (a,\infty)$ which diverges to $\displaystyle \infty$.
    At this point I need to show that the sequence $\displaystyle (g(x_{n}))$ diverges to $\displaystyle \infty$, thereby confirming that the sequence $\displaystyle (f(g(x_{n})))$ converges to L as $\displaystyle n\to \infty$.
    How can I show that the sequence $\displaystyle (g(x_{n}))$ 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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