Задача №2104

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