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Math Help - Simple Limit

  1. #1
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    Exclamation Simple Limit

    Lim (x 0) [(sinx x)/sinx]
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  2. #2
    Super Member Random Variable's Avatar
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    Using L'Hospital's rule,

     \lim_{x \to 0} \frac{\sin x -x}{\sin x} = \lim_{x \to 0} \frac{\cos x -1}{\cos x} = \frac{0}{1} = 0


    or by expanding  \sin x as a Taylor series centered about x=0.

     \lim_{x \to 0} \frac{\sin x-x}{\sin x} = \lim_{x \to 0} \Big(1 - \frac{x}{\sin x}\Big) = \lim_{x \to 0} \Big(1 - \frac{x}{x-\frac{1}{3!}x^{3}+\frac{1}{5!}x^{5}+...}\Big)  = \lim_{x \to 0} \Big(1 - \frac{1}{1-\frac{1}{3!}x^{2}+\frac{1}{5!}x^{4}+...}\Big) = 1-1 =0
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  3. #3
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    Quote Originally Posted by xwrathbringerx View Post
    Lim (x 0) [(sinx – x)/sinx]
    Alternatively to the previous excellent solutions:

    = \lim_{x \rightarrow 0} \left( 1 - \frac{x}{\sin x} \right) = 1 - \lim_{x \rightarrow 0}\frac{x}{\sin x} = 1 -1 = 0

    using a well known standard trig limit.
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