1. **State the problem:** Find the limit as $x$ approaches 8 of the function $f(x) = (3 + \sqrt[3]{x})(2 - 5x^2 + x^3)$. 2. **Recall the limit property:** The limit of a product is the product of the limits, if both limits exist: $$\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$$ 3. **Identify the two functions:** Let $f(x) = 3 + \sqrt[3]{x}$ and $g(x) = 2 - 5x^2 + x^3$. 4. **Evaluate each limit separately:** - For $f(x)$: $$\lim_{x \to 8} (3 + \sqrt[3]{x}) = 3 + \sqrt[3]{8} = 3 + 2 = 5$$ - For $g(x)$: $$\lim_{x \to 8} (2 - 5x^2 + x^3) = 2 - 5(8)^2 + (8)^3 = 2 - 5 \times 64 + 512 = 2 - 320 + 512$$ 5. **Simplify the expression for $g(x)$ limit:** $$2 - 320 + 512 = (2 - 320) + 512 = -318 + 512 = 194$$ 6. **Multiply the two limits:** $$\lim_{x \to 8} f(x)g(x) = 5 \times 194 = 970$$ **Final answer:** $$\boxed{970}$$