1. **State the problem:** Find the limit as $x \to +\infty$ of the rational function $$\frac{3x^5 - 5x + 6}{-6x^5 - 2x^4 + x + 3}.$$\n\n2. **Recall the rule for limits of rational functions at infinity:** When $x$ approaches infinity, the highest degree terms dominate the behavior of the function. So, focus on the terms with the highest power of $x$ in numerator and denominator.\n\n3. **Identify the highest degree terms:** Numerator highest degree term is $3x^5$, denominator highest degree term is $-6x^5$.\n\n4. **Divide numerator and denominator by $x^5$ to simplify:**\n$$\frac{3x^5 - 5x + 6}{-6x^5 - 2x^4 + x + 3} = \frac{\frac{3x^5}{x^5} - \frac{5x}{x^5} + \frac{6}{x^5}}{\frac{-6x^5}{x^5} - \frac{2x^4}{x^5} + \frac{x}{x^5} + \frac{3}{x^5}} = \frac{3 - \frac{5}{x^4} + \frac{6}{x^5}}{-6 - \frac{2}{x} + \frac{1}{x^4} + \frac{3}{x^5}}.$$\n\n5. **Evaluate the limit as $x \to +\infty$:** Terms with $\frac{1}{x^n}$ where $n>0$ approach 0, so\n$$\lim_{x \to +\infty} \frac{3 - \frac{5}{x^4} + \frac{6}{x^5}}{-6 - \frac{2}{x} + \frac{1}{x^4} + \frac{3}{x^5}} = \frac{3 - 0 + 0}{-6 - 0 + 0 + 0} = \frac{3}{-6} = -\frac{1}{2}.$$\n\n**Final answer:** $$\boxed{-\frac{1}{2}}.$$