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Math Help - Need help with l'hopital rule question

  1. #1
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    Need help with l'hopital rule question

    lim (x goes to 0) ( (sinx) / x ) ^ (1/ (x^2) ) = ?
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  2. #2
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    \lim_{x\to 0}\left(\frac{sin(x)}{x}\right)^{\frac{1}{x^{2}}}

    If we let t=sin(x), we get:

    \lim_{t\to 0}\left(\frac{t}{arcsin(t))}\right)^{\frac{1}{arcs  in^{2}(t)}}

    =e^{\left(\lim_{t\to 0}\frac{ln(\frac{t}{arcsin(t)})}{arcsin^{2}(t)}\ri  ght)}

    Using L'Hopital and hammering at it, we get it whittled down to:

    e^{\left(\frac{-1}{6}\lim_{t\to 0}\sqrt{1-t^{2}}\right)}

    =e^{\frac{-1}{6}}
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  3. #3
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    Thanks for the hint, but i still cant get it down to -1/6. I keep getting stuck in the loop of infinity - infinity.


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  4. #4
    Eater of Worlds
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    You have to apply L'Hopital several times.

    Eventually, we get:

    e^{\left(\frac{1}{2}\cdot\frac{\lim_{t\to 0}\sqrt{1-t^{2}}}{\lim_{t\to 0} -3\sqrt{1-t^{2}}+\lim_{t\to 0} tsin^{-1}(t)}\right)}

    Now, you can see it?.

    I hope you can see that. I don't know how to make it bigger.
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  5. #5
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by galactus View Post
    I don't know how to make it bigger.
    This can be achieved using \displaystyle :

    \exp{\displaystyle \left(\frac{1}{2}\cdot\frac{\lim_{t\to 0}\sqrt{1-t^{2}}}{\lim_{t\to 0} -3\sqrt{1-t^{2}}+\lim_{t\to 0} tsin^{-1}(t)}\right)}

    or

    \exp{ \left(\frac{1}{2}\cdot\frac{\displaystyle\lim_{t\t  o 0}\sqrt{1-t^{2}}}{\displaystyle\lim_{t\to 0} -3\sqrt{1-t^{2}}+\lim_{t\to 0} tsin^{-1}(t)}\right)}
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  6. #6
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    I got the answer now. Thanks for the help!!
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