Let's solve the two limit problems from the uploaded image.

Problem 8:

We need to evaluate: $\lim_{x \to 3} \frac{x^2 - 9}{\sqrt{x^2 + 7} - 4}.$

Step-by-step solution:

1. Factorize the numerator: $x^2 - 9 = (x - 3)(x + 3).$

2. Rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator: [ \sqrt{x^2 + 7} - 4 \quad \text{becomes} \quad (\sqrt{x^2 + 7} - 4)(\sqrt{x^2 + 7} +