Calculus

Using Maclaurin expansion of $\sin(3x)$: $\sin(3x) = 3x - \frac{(3x)^3}{3!} + \frac{(3x)^5}{5!} - \dots = 3x - \frac{27 x^3}{6} + O(x^5) = 3x - 4.5 x^3 + O(x^5)$ Substituting into limit: $L = \lim_{x \to 0} \frac{(3x - 4.5 x^3) - 3x}{x^3} = \lim_{x \to 0} \frac{-4.5 x^3}{x^3} = -4.5$