1. **State the problem:** Find the limit as $x$ approaches $-\infty$ of the function $$\frac{2x^4 + 4x^3 - 1}{4x^3 - 3x^5 + 5}.$$\n\n2. **Identify the highest powers:** The numerator's highest power is $x^4$, and the denominator's highest power is $x^5$.\n\n3. **Divide numerator and denominator by the highest power in the denominator, which is $x^5$: $$\frac{\frac{2x^4}{x^5} + \frac{4x^3}{x^5} - \frac{1}{x^5}}{\frac{4x^3}{x^5} - \frac{3x^5}{x^5} + \frac{5}{x^5}} = \frac{2x^{-1} + 4x^{-2} - x^{-5}}{4x^{-2} - 3 + 5x^{-5}}.$$\n\n4. **Evaluate the limit as $x \to -\infty$:** Terms with negative powers of $x$ approach 0, so the expression simplifies to $$\frac{0 + 0 - 0}{0 - 3 + 0} = \frac{0}{-3} = 0.$$\n\n5. **Conclusion:** The limit is 0 as $x$ approaches $-\infty$.