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Math Help - Inverse trig limit

  1. #1
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    Inverse trig limit

    lim x--->1

    arccos ( (1-sqrtx) / (1-x) )

    show all algebaric simplifications

    ty in advance
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  2. #2
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    Quote Originally Posted by am3llcool View Post
    lim x--->1

    arccos ( (1-sqrtx) / (1-x) )

    show all algebaric simplifications

    ty in advance

    Are you sure the limit exists ?

     \lim_{x\to1} \frac{ \cos^{-1}(1-\sqrt{x})}{1-x}

    which has the form  \frac{ \frac{2k+1}{2} \pi}{0}


    I think the question is

     \lim_{x\to1}  \frac{ \sin^{-1}(1-\sqrt{x})}{1-x}


     = \lim_{x\to1}  \frac{ \sin^{-1}(1-\sqrt{x})}{(1-\sqrt{x})(1+\sqrt{x})} = \frac{1}{2}
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  3. #3
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    Quote Originally Posted by am3llcool View Post
    lim x--->1

    arccos ( (1-sqrtx) / (1-x) )

    show all algebaric simplifications
    \lim_{x \to 1} \arccos\left(\frac{1-\sqrt{x}}{(1-\sqrt{x})(1+\sqrt{x})} \right)

    \lim_{x \to 1} \arccos\left(\frac{1}{1+\sqrt{x}}\right) = \frac{\pi}{3}
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