1. **State the problem:** Find the limit $$\lim_{x \to \pi} \frac{\sin x}{2 + \cos x}$$. 2. **Recall the limit rule:** If the function is continuous at the point, the limit is the function value at that point. 3. **Evaluate the numerator at $x=\pi$:** $$\sin \pi = 0$$. 4. **Evaluate the denominator at $x=\pi$:** $$2 + \cos \pi = 2 + (-1) = 1$$. 5. **Substitute these values into the expression:** $$\frac{0}{1} = 0$$. 6. **Conclusion:** The limit is $$0$$ because the denominator is not zero and the numerator approaches zero.