1. **Problem Statement:** Given that $\lim_{x \to a} f(x) = A$ and $\lim_{x \to a} g(x) = B$, we want to understand the rules for limits involving sums, products, quotients, and powers of these functions. 2. **Limit Rules:** The fundamental limit laws are: - Sum/Difference: $$\lim_{x \to a} (f(x) \pm g(x)) = A \pm B$$ - Product: $$\lim_{x \to a} (f(x) \cdot g(x)) = A \cdot B$$ - Quotient (if $B \neq 0$): $$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{A}{B}$$ - Power (if $A^r$ is defined and $r$ is any real number): $$\lim_{x \to a} (f(x))^r = A^r$$ 3. **Explanation:** - These rules allow us to compute limits of combined functions by knowing the limits of individual functions. - For example, if $f(x)$ approaches $A$ and $g(x)$ approaches $B$ as $x$ approaches $a$, then their sum approaches $A+B$. - The quotient rule requires $B \neq 0$ to avoid division by zero. - The power rule requires the limit $A^r$ to be defined (e.g., no negative bases with fractional exponents that are not real). 4. **Special Cases:** - If $f(x)$ is a constant function $c$, then $\lim_{x \to a} f(x) = c$ for all $a$. - If $f(x) = x$, then $\lim_{x \to a} f(x) = a$. 5. **Summary:** These rules simplify limit calculations by breaking complex expressions into simpler parts whose limits are known or easier to find. Final answer: The limit laws stated above hold true and are essential tools in calculus for evaluating limits of combined functions.