1. **Problem Statement:** Evaluate the limit $$\lim_{n \to \infty} \frac{10n^2 + 4n + 1}{7n^4 + 5n}$$. 2. **Formula and Rules:** When evaluating limits at infinity for rational functions, divide numerator and denominator by the highest power of $n$ in the denominator to simplify. 3. **Work:** The highest power in the denominator is $n^4$. $$\lim_{n \to \infty} \frac{10n^2 + 4n + 1}{7n^4 + 5n} = \lim_{n \to \infty} \frac{\frac{10n^2}{n^4} + \frac{4n}{n^4} + \frac{1}{n^4}}{\frac{7n^4}{n^4} + \frac{5n}{n^4}} = \lim_{n \to \infty} \frac{10n^{-2} + 4n^{-3} + n^{-4}}{7 + 5n^{-3}}$$ 4. **Evaluate:** As $n \to \infty$, terms with negative powers of $n$ approach 0. So the limit becomes: $$\frac{0 + 0 + 0}{7 + 0} = \frac{0}{7} = 0$$ 5. **Answer:** The limit is 0. --- 1. **Problem Statement:** Evaluate the limit $$\lim_{k \to \infty} \frac{-6k^2 - 5k + 3}{5k^6 + 3k}$$. 2. **Formula and Rules:** Use the same approach: divide numerator and denominator by the highest power of $k$ in the denominator. 3. **Work:** The highest power in the denominator is $k^6$. $$\lim_{k \to \infty} \frac{-6k^2 - 5k + 3}{5k^6 + 3k} = \lim_{k \to \infty} \frac{\frac{-6k^2}{k^6} + \frac{-5k}{k^6} + \frac{3}{k^6}}{\frac{5k^6}{k^6} + \frac{3k}{k^6}} = \lim_{k \to \infty} \frac{-6k^{-4} - 5k^{-5} + 3k^{-6}}{5 + 3k^{-5}}$$ 4. **Evaluate:** As $k \to \infty$, terms with negative powers of $k$ approach 0. So the limit becomes: $$\frac{0 - 0 + 0}{5 + 0} = \frac{0}{5} = 0$$ 5. **Answer:** The limit is 0.