Задача №2103

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