To solve the given limit:

$\lim_{x \to 7} \frac{-2x - 2}{x^2 - 1}$

Step 1: Substitution

Substituting $x = 7$ directly:

$\text{Numerator: } -2(7) - 2 = -14 - 2 = -16$ $\text{Denominator: } 7^2 - 1 = 49 - 1 = 48$

Thus, the limit is:

$\frac{-16}{48} = \frac{-1}{3}$

Final Answer:

$\lim_{x \to 7} \frac{-2x - 2}{x^2 - 1} = \boxed{-\frac{1}{3}}$

Let me know if you want details about this solution!

Here are 5 relative questions to expand your understanding:

1. What happens if the numerator or denominator becomes $0$ when evaluating a limit?
2. How do you handle indeterminate forms such as $\frac{0}{0}$?
3. How can limits be solved using factoring or simplification?
4. How does substitution work in evaluating limits of rational functions?
5. Can L'Hôpital's Rule be applied to this problem? Why or why not?

Tip: Always check if substitution works first when solving limits—it’s the simplest method.