1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{6 - 3x + 10x^2}{-2x^4 + 7x^3 + 1}$$ as $x$ approaches 1. 2. **Substitute $x=1$ directly:** $$\frac{6 - 3(1) + 10(1)^2}{-2(1)^4 + 7(1)^3 + 1} = \frac{6 - 3 + 10}{-2 + 7 + 1} = \frac{13}{6}$$ 3. Since direct substitution does not produce an indeterminate form, the limit is simply the value of the function at $x=1$. 4. **Final answer:** $$\lim_{x \to 1} \frac{6 - 3x + 10x^2}{-2x^4 + 7x^3 + 1} = \frac{13}{6}$$