1. **Problem statement:** Given $\lim_{x \to c} f(x) = 10$ and $\lim_{x \to c} g(x) = -14$, evaluate the following limits using limit laws. 2. **Limit laws used:** - Sum law: $\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$ - Difference law: $\lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$ - Product law: $\lim_{x \to c} [f(x) g(x)] = \lim_{x \to c} f(x) \times \lim_{x \to c} g(x)$ - Quotient law: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, provided $\lim_{x \to c} g(x) \neq 0$ 3. **Calculations:** 1. $\lim_{x \to c} (f(x) + 8g(x)) = \lim_{x \to c} f(x) + 8 \lim_{x \to c} g(x) = 10 + 8 \times (-14) = 10 - 112 = -102$ 2. $\lim_{x \to c} (f(x) - g(x)) = \lim_{x \to c} f(x) - \lim_{x \to c} g(x) = 10 - (-14) = 10 + 14 = 24$ 3. $\lim_{x \to c} (f(x) g(x)) = \lim_{x \to c} f(x) \times \lim_{x \to c} g(x) = 10 \times (-14) = -140$ 4. $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} = \frac{10}{-14} = -\frac{5}{7}$ 5. $\lim_{x \to c} \frac{g(x)}{f(x)} = \frac{\lim_{x \to c} g(x)}{\lim_{x \to c} f(x)} = \frac{-14}{10} = -\frac{7}{5}$ **Final answers:** 1. $-102$ 2. $24$ 3. $-140$ 4. $-\frac{5}{7}$ 5. $-\frac{7}{5}$