1. **Problem:** Compute $\lim_{x \to c} (f(x) + g(x))$ given $\lim_{x \to c} f(x) = 1$ and $\lim_{x \to c} g(x) = -1$. 2. **Formula:** The limit of a sum is the sum of the limits: $$\lim_{x \to c} (f(x) + g(x)) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$$ 3. **Calculation:** $$\lim_{x \to c} (f(x) + g(x)) = 1 + (-1) = 0$$ --- 1. **Problem:** Compute $\lim_{x \to c} (f(x) - g(x) - h(x))$ given $\lim_{x \to c} f(x) = 1$, $\lim_{x \to c} g(x) = -1$, and $\lim_{x \to c} h(x) = 2$. 2. **Formula:** The limit of a difference is the difference of the limits: $$\lim_{x \to c} (f(x) - g(x) - h(x)) = \lim_{x \to c} f(x) - \lim_{x \to c} g(x) - \lim_{x \to c} h(x)$$ 3. **Calculation:** $$\lim_{x \to c} (f(x) - g(x) - h(x)) = 1 - (-1) - 2 = 1 + 1 - 2 = 0$$ --- 1. **Problem:** Compute $\lim_{x \to c} (3 \cdot g(x) + 5 \cdot h(x))$. 2. **Formula:** Limits respect scalar multiplication and addition: $$\lim_{x \to c} (a \cdot f(x) + b \cdot g(x)) = a \cdot \lim_{x \to c} f(x) + b \cdot \lim_{x \to c} g(x)$$ 3. **Calculation:** $$\lim_{x \to c} (3 \cdot g(x) + 5 \cdot h(x)) = 3 \cdot (-1) + 5 \cdot 2 = -3 + 10 = 7$$ --- 1. **Problem:** Compute $\lim_{x \to c} \sqrt{f(x)}$. 2. **Formula:** The limit of a continuous function of $f(x)$ is the function of the limit: $$\lim_{x \to c} \sqrt{f(x)} = \sqrt{\lim_{x \to c} f(x)}$$ 3. **Calculation:** $$\lim_{x \to c} \sqrt{f(x)} = \sqrt{1} = 1$$ --- 1. **Problem:** Compute $\lim_{x \to c} (h(x))^5$. 2. **Formula:** The limit of a power is the power of the limit: $$\lim_{x \to c} (h(x))^5 = (\lim_{x \to c} h(x))^5$$ 3. **Calculation:** $$\lim_{x \to c} (h(x))^5 = 2^5 = 32$$ --- 1. **Problem:** Compute $\lim_{x \to c} \frac{g(x) - h(x)}{f(x)}$. 2. **Formula:** The limit of a quotient is the quotient of the limits (if denominator limit is not zero): $$\lim_{x \to c} \frac{g(x) - h(x)}{f(x)} = \frac{\lim_{x \to c} g(x) - \lim_{x \to c} h(x)}{\lim_{x \to c} f(x)}$$ 3. **Calculation:** $$\lim_{x \to c} \frac{g(x) - h(x)}{f(x)} = \frac{-1 - 2}{1} = \frac{-3}{1} = -3$$ --- 1. **Problem:** Compute $\lim_{x \to c} \frac{1}{g(x) + h(x)}$. 2. **Formula:** The limit of a reciprocal is the reciprocal of the limit (if denominator limit is not zero): $$\lim_{x \to c} \frac{1}{g(x) + h(x)} = \frac{1}{\lim_{x \to c} g(x) + \lim_{x \to c} h(x)}$$ 3. **Calculation:** $$\lim_{x \to c} \frac{1}{g(x) + h(x)} = \frac{1}{-1 + 2} = \frac{1}{1} = 1$$