1. Stating the problem: Find the limit as $x$ approaches infinity of the expression $$\frac{7x^4 - x^6}{9x^6 + 9x^8}$$. 2. Formula and rules: When finding limits at infinity for rational functions, divide numerator and denominator by the highest power of $x$ present in the denominator to simplify. 3. Identify the highest power of $x$ in the denominator: $x^8$. 4. Divide numerator and denominator by $x^8$: $$\frac{\frac{7x^4}{x^8} - \frac{x^6}{x^8}}{\frac{9x^6}{x^8} + \frac{9x^8}{x^8}} = \frac{7x^{4-8} - x^{6-8}}{9x^{6-8} + 9x^{8-8}} = \frac{7x^{-4} - x^{-2}}{9x^{-2} + 9}$$ 5. Simplify the expression: $$\frac{7x^{-4} - x^{-2}}{9x^{-2} + 9} = \frac{\frac{7}{x^4} - \frac{1}{x^2}}{\frac{9}{x^2} + 9}$$ 6. Evaluate the limit as $x \to \infty$: Since $\lim_{x \to \infty} \frac{1}{x^n} = 0$ for any positive $n$, we have: $$\lim_{x \to \infty} \frac{\frac{7}{x^4} - \frac{1}{x^2}}{\frac{9}{x^2} + 9} = \frac{0 - 0}{0 + 9} = \frac{0}{9} = 0$$ Final answer: $$\boxed{0}$$