The value of $$\lim_{x \to 0}\left(\frac{x^2 \sin^2 x}{x^2 - \sin^2 x}\right)$$ is:

$$L = \lim_{x \to 0} \frac{x^2 \sin^2 x}{x^2 - \sin^2 x}$$

$$L = \lim_{x \to 0} \frac{x^2 \sin^2 x}{(x - \sin x)(x + \sin x)}$$

$$L = \lim_{x \to 0} \left( \frac{\sin x}{x} \right)^2 \cdot \frac{x^4}{(x - \sin x)(x + \sin x)}$$

$$L = \lim_{x \to 0} \left( \frac{\sin x}{x} \right)^2 \cdot \left( \frac{x^3}{x - \sin x} \right) \cdot \left( \frac{x}{x + \sin x} \right)$$

$$L = (1)^2 \cdot \left( \frac{1}{1/6} \right) \cdot \left( \frac{1}{1 + 1} \right)$$

$$L = 1 \cdot 6 \cdot \frac{1}{2}$$

$$L = 3$$