1. We are asked to evaluate the limit as $n \to 0$ of the expression $$\int_0^\pi (\sin \theta - \sin \theta) \lim_{n\to 0} \frac{\sqrt{n+\theta} - \sqrt{n+\theta}}{\theta} d\theta$$ 2. Inside the integral, the term $(\sin \theta - \sin \theta)$ simplifies to 0 for all $\theta$ since they are identical terms: $$\sin \theta - \sin \theta = 0$$ 3. Also, the numerator in the limit fraction is $$\sqrt{n+\theta} - \sqrt{n+\theta} = 0$$ so the entire fraction is $0/\theta = 0$ for each $\theta \neq 0$. 4. Since the integrand is 0 for all $\theta$ in $[0, \pi]$, the integral over this interval is: $$\int_0^\pi 0 \ d\theta = 0$$ 5. Therefore, the limit as $n \to 0$ does not affect this result and the value of the entire expression is $$0$$ Final answer: $0$