1. **Problem:** Compute $$\lim_{x \to a} \frac{3f(x) + g(x)}{h(x)}$$ given $$\lim_{x \to a} f(x) = 5$$, $$\lim_{x \to a} g(x) = -3$$, and $$\lim_{x \to a} h(x) = -2$$. 2. **Formula and rules:** The limit of a quotient is the quotient of the limits, provided the denominator's limit is not zero: $$\lim_{x \to a} \frac{u(x)}{v(x)} = \frac{\lim_{x \to a} u(x)}{\lim_{x \to a} v(x)}$$ if $$\lim_{x \to a} v(x) \neq 0$$. 3. **Intermediate work:** Calculate numerator limit: $$\lim_{x \to a} (3f(x) + g(x)) = 3 \cdot \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = 3 \cdot 5 + (-3) = 15 - 3 = 12$$ Calculate denominator limit: $$\lim_{x \to a} h(x) = -2$$ 4. **Evaluate limit:** $$\lim_{x \to a} \frac{3f(x) + g(x)}{h(x)} = \frac{12}{-2} = -6$$ --- 1. **Problem:** Compute $$\lim_{x \to a} \{ [f(x)]^2 [g(x)]^3 - 7h(x) \}$$. 2. **Formula and rules:** The limit of sums and products is the sum and product of the limits: $$\lim_{x \to a} (u(x) \pm v(x)) = \lim_{x \to a} u(x) \pm \lim_{x \to a} v(x)$$ $$\lim_{x \to a} (u(x) v(x)) = \lim_{x \to a} u(x) \cdot \lim_{x \to a} v(x)$$ 3. **Intermediate work:** Calculate each limit: $$\lim_{x \to a} [f(x)]^2 = (\lim_{x \to a} f(x))^2 = 5^2 = 25$$ $$\lim_{x \to a} [g(x)]^3 = (\lim_{x \to a} g(x))^3 = (-3)^3 = -27$$ $$\lim_{x \to a} h(x) = -2$$ Calculate product and subtraction: $$25 \times (-27) - 7 \times (-2) = -675 + 14 = -661$$ 4. **Evaluate limit:** $$\lim_{x \to a} \{ [f(x)]^2 [g(x)]^3 - 7h(x) \} = -661$$ --- 1. **Problem:** Compute $$\lim_{x \to a} \frac{\sqrt{-3 f(x) g(x)}}{h(x)}$$. 2. **Formula and rules:** Limits of roots and quotients: $$\lim_{x \to a} \sqrt{u(x)} = \sqrt{\lim_{x \to a} u(x)}$$ if the limit inside the root is non-negative. 3. **Intermediate work:** Calculate inside the root: $$\lim_{x \to a} (-3 f(x) g(x)) = -3 \times \lim_{x \to a} f(x) \times \lim_{x \to a} g(x) = -3 \times 5 \times (-3) = 45$$ Since 45 is positive, root is defined. Calculate denominator limit: $$\lim_{x \to a} h(x) = -2$$ 4. **Evaluate limit:** $$\lim_{x \to a} \frac{\sqrt{-3 f(x) g(x)}}{h(x)} = \frac{\sqrt{45}}{-2} = \frac{3 \sqrt{5}}{-2} = -\frac{3 \sqrt{5}}{2}$$ **Final answers:** 1. $$-6$$ 2. $$-661$$ 3. $$-\frac{3 \sqrt{5}}{2}$$