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Thread: Csc Limit question

  1. #1
    Senior Member polymerase's Avatar
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    Csc Limit question

    Find $\displaystyle f'(\frac{\pi}{4})$ if $\displaystyle f(x)=\lim_{t\to{x}}\frac{csc\;t -csc\;x}{t-x}$
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by polymerase View Post
    Find $\displaystyle f'(\frac{\pi}{4})$ if $\displaystyle f(x)=\lim_{t\to{x}}\frac{csc\;t -csc\;x}{t-x}$
    note that $\displaystyle f(x)$ is defined by the limit definition of the derivative of $\displaystyle \csc x$

    thus, let $\displaystyle g(x) = \csc x$

    we have that $\displaystyle f(x) = g'(x)$

    so $\displaystyle f'(x) = g''(x)$

    therefore, $\displaystyle f' \left( \frac {\pi}4 \right) = g'' \left( \frac {\pi}4 \right) $

    now find $\displaystyle g''(x)$ and you will have your answer


    are you required to find the derivative by evaluating the limit? i doubt it. it will especially be a pain doing it for the second derivative
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  3. #3
    Senior Member polymerase's Avatar
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    Quote Originally Posted by Jhevon View Post
    note that $\displaystyle f(x)$ is defined by the limit definition of the derivative of $\displaystyle \csc x$

    thus, let $\displaystyle g(x) = \csc x$

    we have that $\displaystyle f(x) = g'(x)$

    so $\displaystyle f'(x) = g''(x)$

    therefore, $\displaystyle f' \left( \frac {\pi}4 \right) = g'' \left( \frac {\pi}4 \right) $

    now find $\displaystyle g''(x)$ and you will have your answer


    are you required to find the derivative by evaluating the limit? i doubt it. it will especially be a pain doing it for the second derivative
    maybe im just doing it wrong but i'm getting $\displaystyle \sqrt{2}+\sqrt{8}$ and thats not right
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