1. **Stating the problem:** Evaluate the expression involving an integral and a limit: $$\int_s^x 3 \lim_{s \to x} \frac{3! + 5!}{2 \pi i} \, dt$$ 2. **Understanding the components:** - The limit is taken as $s$ approaches $x$ of the constant fraction $\frac{3! + 5!}{2 \pi i}$. - The factorials are $3! = 6$ and $5! = 120$. - The integral is with respect to $t$ from $s$ to $x$ of the constant value $3$ times the limit. 3. **Calculate the limit:** Since the expression inside the limit does not depend on $s$, the limit is simply the value itself: $$\lim_{s \to x} \frac{3! + 5!}{2 \pi i} = \frac{6 + 120}{2 \pi i} = \frac{126}{2 \pi i} = \frac{63}{\pi i}$$ 4. **Rewrite the integral:** The integral becomes: $$\int_s^x 3 \cdot \frac{63}{\pi i} \, dt = \int_s^x \frac{189}{\pi i} \, dt$$ 5. **Evaluate the integral:** Since the integrand is constant with respect to $t$: $$\int_s^x \frac{189}{\pi i} \, dt = \frac{189}{\pi i} (x - s)$$ 6. **Final answer:** $$\boxed{\frac{189}{\pi i} (x - s)}$$ This is the evaluated value of the given expression.