\sin x의 매클로린 급수 \sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots 를 이용한다.

\displaystyle\lim_{x \to 0} \dfrac{x - \sin x}{x^3} = \displaystyle\lim_{x \to 0} \dfrac{x - \bigg(x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots\bigg)}{x^3} = \displaystyle\lim_{x \to 0} \dfrac{\dfrac{x^3}{3!} - \dfrac{x^5}{5!} + \cdots}{x^3}=\displaystyle\lim_{x \to 0} \bigg(\dfrac{1}{6} - \dfrac{x^2}{120} + \cdots\bigg) = \dfrac{1}{6}