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I've seen lots of questions demonstrating how to prove a limit doesn't exist, however there aren't any solutions using composite functions and limits at infinity. The question I'm trying to solve is as follows:

Let f: [-1, 1] --> R be a function. Suppose that f is not a constant function. Define g: R --> R by g(x) = f(sin(x)). Prove that limx→∞ g(x) does not exist.

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As $f$ is non-constant, it will take at least two different values. Then $g$ alternates periodically between them and cannot converge.

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  • $\begingroup$ Thanks for the quick response. I'm not sure that this is enough to complete the proof. What I'm trying to do right now is do separate epsilon proofs for g(x) and sin(x), and then try to come to a contradiction. $\endgroup$ Jan 4 '20 at 16:12
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    $\begingroup$ @S.Miller: take $\epsilon$ smaller than half the difference between the values mentioned in my answer. $\endgroup$
    – user65203
    Jan 4 '20 at 16:15

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