1. **State the problem:** Find the limit $$\lim_{x \to \pi} \frac{\sin x}{\pi - x}$$. 2. **Recall the formula and important rule:** This is a limit of the form $$\frac{\sin x}{\pi - x}$$ as $$x$$ approaches $$\pi$$. Notice that $$\sin \pi = 0$$, so the numerator approaches 0 and the denominator approaches 0, which is an indeterminate form $$\frac{0}{0}$$. 3. **Use substitution:** Let $$t = \pi - x$$. Then as $$x \to \pi$$, $$t \to 0$$. 4. **Rewrite the limit in terms of $$t$$:** $$ \lim_{x \to \pi} \frac{\sin x}{\pi - x} = \lim_{t \to 0} \frac{\sin(\pi - t)}{t} $$ 5. **Use the identity:** $$\sin(\pi - t) = \sin t$$. 6. **Simplify the limit:** $$ \lim_{t \to 0} \frac{\sin t}{t} $$ 7. **Recall the standard limit:** $$ \lim_{t \to 0} \frac{\sin t}{t} = 1 $$ 8. **Final answer:** $$ \boxed{1} $$