1. **State the problem:** Find the limit of the sequence $$u_n = \frac{1 - n^3}{n - 5n^4}$$ as $n$ approaches infinity. 2. **Recall the formula and rules:** When finding limits of sequences involving polynomials, the highest power terms dominate the behavior as $n \to \infty$. 3. **Analyze the expression:** $$u_n = \frac{1 - n^3}{n - 5n^4}$$ 4. **Divide numerator and denominator by the highest power of $n$ in the denominator, which is $n^4$:** $$u_n = \frac{\frac{1}{n^4} - \frac{n^3}{n^4}}{\frac{n}{n^4} - \frac{5n^4}{n^4}} = \frac{\frac{1}{n^4} - \frac{1}{n}}{\frac{1}{n^3} - 5}$$ 5. **Evaluate the limit as $n \to \infty$:** - $\frac{1}{n^4} \to 0$ - $\frac{1}{n} \to 0$ - $\frac{1}{n^3} \to 0$ So, $$\lim_{n \to \infty} u_n = \frac{0 - 0}{0 - 5} = \frac{0}{-5} = 0$$ 6. **Conclusion:** The limit of the sequence is 0.