1. The problem asks to find the limit $$\lim_{x \to \pi} \frac{2 \sin x}{x - \pi}.$$\n\n2. Observe that this is a limit which resembles the form $$\lim_{x \to a} \frac{f(x)}{x - a}$$ where $$f(x) = 2 \sin x$$ and $$a = \pi.$$\n\n3. Direct substitution gives $$\frac{2 \sin \pi}{\pi - \pi} = \frac{0}{0}$$ which is indeterminate, so we can apply L'Hôpital's rule or rewrite the limit.\n\n4. Using L'Hôpital's rule, differentiate numerator and denominator with respect to $$x$$:\n$$\text{Numerator derivative} = 2 \cos x$$\n$$\text{Denominator derivative} = 1.$$\n\n5. Then the limit becomes $$\lim_{x \to \pi} 2 \cos x = 2 \cos \pi = 2 \cdot (-1) = -2.$$\n\n6. Thus, the value of the limit is $$\boxed{-2}.$$