Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 106: 22

$$\frac{1}{3}$$

Given $$\lim_{x\to 0 }\frac{\sin^2 x}{3x^2}$$ Then \begin{align*} \lim_{x\to 0 }\frac{\sin^2 x}{3x^2}&=\frac{1}{3}\lim_{x\to 0 }\frac{\sin^2 x}{ x^2}\\ &=\frac{1}{3}\left(\lim_{x\to 0 }\frac{\sin x}{ x } \right)^2 \\ &=\frac{1}{3} \end{align*}