1. **State the problem:** Find the limit $$\lim_{x \to 7} \frac{x^2 - 9}{x - 7}$$. 2. **Identify the issue:** Direct substitution gives $$\frac{7^2 - 9}{7 - 7} = \frac{49 - 9}{0} = \frac{40}{0}$$ which is undefined. So, we need to simplify the expression. 3. **Factor the numerator:** $$x^2 - 9$$ is a difference of squares, so $$x^2 - 9 = (x - 3)(x + 3)$$. 4. **Rewrite the limit:** $$\lim_{x \to 7} \frac{(x - 3)(x + 3)}{x - 7}$$. 5. **Since the denominator is $$x - 7$$ and numerator has no $$x - 7$$ factor, the limit tends to infinity or negative infinity depending on the sign near 7. But let's check the behavior: 6. **Evaluate numerator and denominator near 7:** - Numerator at 7: $$(7 - 3)(7 + 3) = 4 \times 10 = 40$$ - Denominator near 7: approaches 0. 7. **Conclusion:** The limit does not exist as a finite number because the denominator approaches zero while numerator approaches 40. The function tends to infinity or negative infinity depending on the direction. **Final answer:** The limit does not exist (it diverges).