1. **State the problem:** Show, using the properties of limits, that if $$\lim_{x \to 5} f(x) = 3,$$ then $$\lim_{x \to 5} \frac{x^2 - 4}{f(x)} = 7.$$ 2. **Recall the properties of limits:** - The limit of a quotient is the quotient of the limits, provided the denominator's limit is not zero. - The limit of a sum, difference, product, or power is the sum, difference, product, or power of the limits. 3. **Evaluate the numerator limit:** $$\lim_{x \to 5} (x^2 - 4) = (5)^2 - 4 = 25 - 4 = 21.$$ 4. **Evaluate the denominator limit:** Given $$\lim_{x \to 5} f(x) = 3.$$ 5. **Apply the quotient limit property:** $$\lim_{x \to 5} \frac{x^2 - 4}{f(x)} = \frac{\lim_{x \to 5} (x^2 - 4)}{\lim_{x \to 5} f(x)} = \frac{21}{3} = 7.$$ 6. **Conclusion:** Therefore, $$\lim_{x \to 5} \frac{x^2 - 4}{f(x)} = 7,$$ as required.