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Thread: limit of a function

  1. #1
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    limit of a function

    hello,i tried to prove that the following limit in +and-infinity is equal to 0 but i couldn't,how can i prove that the limit of f(sin(x)) is a real number?thank you in advance for any help.(f is continuous in R)
    limit of a function-bigiig.gif
    Last edited by aray; Sep 4th 2017 at 07:19 AM.
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  2. #2
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    Re: limit of a function

    For all real $x $, $-1 \le \sin x \le 1$. By the extreme value theorem, a continuous function achieves a maximum and a minimum value over a closed interval, so there exists $c_1,c_2$ such that for all $x \in \mathbb{R} $, $f(\sin c_1) \le f(\sin x) \le f(\sin c_2)$.

    Now you can use the Squeeze Theorem.
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  3. #3
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    Re: limit of a function

    It's worth pointing out that the extreme value theorem can be used only because both f and \sin x are continuous, and thus f(\sin x) is also continuous.
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  4. #4
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    Re: limit of a function

    ok,thank you for your help
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