5. Вычислите: 1) lim \( \frac{-7n^4+6n^2-1}{6-n+8n^4} \); 2) lim \( \frac{2n^2-1}{5+n^2} \).

Решение:

1. 1)

\[\lim_{n \to \infty} \frac{-7n^4+6n^2-1}{6-n+8n^4} = \lim_{n \to \infty} \frac{n^4(-7+\frac{6}{n^2}-\frac{1}{n^4})}{n^4(\frac{6}{n^4}-\frac{1}{n^3}+8)} = \frac{-7+0-0}{0-0+8} = -\frac{7}{8}\]

2. 2)

\[\lim_{n \to \infty} \frac{2n^2-1}{5+n^2} = \lim_{n \to \infty} \frac{n^2(2-\frac{1}{n^2})}{n^2(\frac{5}{n^2}+1)} = \frac{2-0}{0+1} = 2\]