Home » Blog » Find the limit as x goes to 0 of (sin x – x) / x3 Find the limit as x goes to 0 of (sin x – x) / x3 by RoRi December 21, 2015 Evaluate the limit. Using the expansion (p. 287) we compute, Related
![\[ \lim_{x \to 0} \frac{\sin x - x}{x^3}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-3daefd39ab8a5ccd47ebf46b6de5679a_l3.png "Rendered by QuickLaTeX.com")
![\[ \sin x = x - \frac{x^3}{3!} + o(x^4) \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-7b8927720f85124d3f36d5c970e703db_l3.png "Rendered by QuickLaTeX.com")
![\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*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-7fcf178c193e2ebf7a64e1258befb1b0_l3.png "Rendered by QuickLaTeX.com")