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Find the limit as x goes to 0 of (sin x – x) / x3

Evaluate the limit.

    \[ \lim_{x \to 0} \frac{\sin x - x}{x^3}. \]


Using the expansion (p. 287)

    \[ \sin x = x - \frac{x^3}{3!} + o(x^4) \]

we compute,

    \begin{align*}  \lim_{x \to 0} \frac{\sin x - x}{x^3} &= \lim_{x \to 0} \frac{x - \frac{x^3}{6} + o(x^4) - x}{x^3} \\[9pt]  &= \lim_{x \to 0} \frac{-1 + o(x)}{6} \\[9pt]  &= \lim_{x \to 0} \left(\frac{-1}{6} + o(x)\right) \\[9pt]  &= -\frac{1}{6}. \end{align*}

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