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Math Help - Limits problem

  1. #1
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    Limits problem

    find limits

    lim tanx - x.............
    x->0...sin(2x) - sinx - x
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  2. #2
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    Quote Originally Posted by manalive04 View Post
    find limits

    lim tanx - x.............
    x->0...sin(2x) - sinx - x
    Try L'Hopital's rule.
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  3. #3
    Super Member Random Variable's Avatar
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     \lim_{x \to 0} \ \frac{\tan x -x}{\sin 2x -\sin x - x}

     = \lim_{x \to 0} \ \frac { \Big(x + \frac{1}{3}x^{3} + \frac{2}{15} x^{5} + ... \Big) - x } { \Big( 2x - \frac{8}{3!}x^{3} + \frac{32}{5!} x^{5} + ... \Big) - \Big(x -\frac{1}{3!}x^{3} + \frac{1}{5!}x^{5}+ ...\big) -x}

     = \lim_{x \to 0} \ \frac { \frac{1}{3}x^{3} + \frac{2}{15} x^{5} + ... }{\frac{-7}{3!}x^{3} + \frac{31}{5!}x^{5} + ...} = \lim_{x \to 0} \ \frac { \frac{1}{3} + \frac{2}{15} x^{2} + ... }{\frac{-7}{3!}+ \frac{31}{5!}x^{2} + ...} = \frac{\frac{1}{3}}{\frac{-7}{3!}}= \frac{-2}{7}
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