1. **Problem:** Calculate the limit $$\lim_{x \to 5} \frac{x^2 - 2x - 15}{x^2 - 25}$$ 2. **Recall the formula:** Limits involving rational functions can often be simplified by factoring numerator and denominator and canceling common factors. 3. **Factor numerator and denominator:** $$x^2 - 2x - 15 = (x - 5)(x + 3)$$ $$x^2 - 25 = (x - 5)(x + 5)$$ 4. **Rewrite the limit:** $$\lim_{x \to 5} \frac{(x - 5)(x + 3)}{(x - 5)(x + 5)}$$ 5. **Cancel common factor $(x - 5)$:** $$\lim_{x \to 5} \frac{\cancel{(x - 5)}(x + 3)}{\cancel{(x - 5)}(x + 5)} = \lim_{x \to 5} \frac{x + 3}{x + 5}$$ 6. **Evaluate the limit by direct substitution:** $$\frac{5 + 3}{5 + 5} = \frac{8}{10} = \frac{4}{5}$$ 7. **Answer:** $$\boxed{\frac{4}{5}}$$