1. **State the problem:** Compute the limit $$\lim_{x \to \infty} \frac{5x^4 - 7x + 2}{-2x^4 + 2x - 3}$$. 2. **Recall the rule for limits of rational functions as $x \to \infty$:** When the degrees of the numerator and denominator polynomials are the same, the limit is the ratio of the leading coefficients. 3. **Identify the leading terms:** - Numerator leading term: $5x^4$ - Denominator leading term: $-2x^4$ 4. **Divide numerator and denominator by $x^4$ to simplify:** $$\lim_{x \to \infty} \frac{5x^4 - 7x + 2}{-2x^4 + 2x - 3} = \lim_{x \to \infty} \frac{5 - \frac{7}{x^3} + \frac{2}{x^4}}{-2 + \frac{2}{x^3} - \frac{3}{x^4}}$$ 5. **Evaluate the limit as $x \to \infty$:** Terms with $\frac{1}{x^n}$ go to zero, so $$= \frac{5 - 0 + 0}{-2 + 0 - 0} = \frac{5}{-2} = -\frac{5}{2}$$ **Final answer:** $$\boxed{-\frac{5}{2}}$$