1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \frac{4x^2 + 20x}{5x^2 + 18}$$. 2. **Recall the rule for limits of rational functions:** When $x$ approaches infinity, the behavior of a rational function $$\frac{P(x)}{Q(x)}$$ where $P$ and $Q$ are polynomials is dominated by the highest degree terms. 3. **Identify the highest degree terms:** In numerator, highest degree term is $4x^2$; in denominator, it is $5x^2$. 4. **Divide numerator and denominator by $x^2$ (the highest power):** $$\lim_{x \to \infty} \frac{4x^2 + 20x}{5x^2 + 18} = \lim_{x \to \infty} \frac{\frac{4x^2}{x^2} + \frac{20x}{x^2}}{\frac{5x^2}{x^2} + \frac{18}{x^2}} = \lim_{x \to \infty} \frac{4 + \frac{20}{x}}{5 + \frac{18}{x^2}}$$ 5. **Simplify the fractions:** $$\lim_{x \to \infty} \frac{4 + \frac{20}{x}}{5 + \frac{18}{x^2}}$$ 6. **Evaluate the limit as $x \to \infty$:** Terms with $\frac{1}{x}$ and $\frac{1}{x^2}$ approach 0, so $$\lim_{x \to \infty} \frac{4 + 0}{5 + 0} = \frac{4}{5}$$. **Final answer:** $$\boxed{\frac{4}{5}}$$