$$\begin{aligned} & \lim _\limits{x \rightarrow 0} f(x)=f(0) \quad \text { (continuous at } x=0) \\ & \lim _{x \rightarrow 0} \frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3} \end{aligned}$$

For limit to exist $$\beta=0$$

$$\begin{aligned} & \Rightarrow \lim _{x \rightarrow 0} \frac{\sin 3 x+\alpha \sin x}{x^3} \\ & \Rightarrow \lim _{x \rightarrow 0} \frac{(3+\alpha) \sin x-4 \sin ^3 x}{x^3} \end{aligned}$$

For limit to exist $$\alpha+3=0 \Rightarrow \alpha=-3$$

$$\Rightarrow \lim _\limits{x \rightarrow 0} \frac{-4 \sin ^3 x}{x^3}=-4=f(0)$$