1. **State the problem:** Find the horizontal asymptotes of the function $$f(x) = \frac{5x^{2} - 10}{10x^{2} - 12x + 20}$$. 2. **Recall the rule for horizontal asymptotes:** For rational functions $$\frac{P(x)}{Q(x)}$$ where $$P(x)$$ and $$Q(x)$$ are polynomials, the horizontal asymptote depends on the degrees of $$P(x)$$ and $$Q(x)$$: - If degree of numerator = degree of denominator, horizontal asymptote is $$y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$$. - If degree numerator < degree denominator, horizontal asymptote is $$y=0$$. - If degree numerator > degree denominator, no horizontal asymptote. 3. **Identify degrees:** Both numerator and denominator are degree 2. 4. **Find leading coefficients:** Numerator leading coefficient is 5, denominator leading coefficient is 10. 5. **Calculate horizontal asymptote:** $$y = \frac{5}{10} = \frac{1}{2}$$ 6. **Conclusion:** The horizontal asymptote is $$y = \frac{1}{2}$$. 7. **Check options:** a. $$f(x) = \frac{1}{2}$$ (correct horizontal asymptote) b. $$x=2$$ (vertical line, not horizontal asymptote) c. $$x=5$$ (vertical line, not horizontal asymptote) d. Do not exist (incorrect) **Final answer:** a. $$f(x) = \frac{1}{2}$$