1. **State the problem:** Find the horizontal asymptote(s) of the function $$f(x) = \frac{3x^4 + 2x + 2}{5x^4 + 2x - 4}$$. 2. **Recall the rule for horizontal asymptotes of rational functions:** - If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $$y=0$$. - If the degrees are equal, the horizontal asymptote is $$y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$$. - If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. 3. **Identify degrees:** - Degree of numerator = 4 (highest power of $$x$$ in numerator is $$x^4$$). - Degree of denominator = 4. 4. **Since degrees are equal, horizontal asymptote is:** $$y = \frac{3}{5}$$ 5. **No other horizontal asymptotes exist because the degrees are equal and the function behaves like the ratio of leading terms at infinity.** **Final answer:** The function has one horizontal asymptote, $$y=\frac{3}{5}$$.