Задача №2112

\[ \lim_{n\to\infty}\frac{(n+2)^4-(n-2)^4}{(n+5)^2+(n-5)^2} =\lim_{n\to\infty}\frac{n^4+8n^3+24n^2+32n+16-\left(n^4-8n^3+24n^2-32n+16\right)}{(n+5)^2+(n-5)^2}=\\ =\lim_{n\to\infty}\frac{16n^3+64n}{(n+5)^2+(n-5)^2} =\left[\frac{\infty}{\infty}\right] =\lim_{n\to\infty}\frac{\frac{16n^3+64n}{n^3}}{\frac{(n+5)^2+(n-5)^2}{n^3}} =\lim_{n\to\infty}\frac{16+\frac{64}{n^2}}{\frac{1}{n}\cdot\left(1+\frac{5}{n}\right)^2+\frac{1}{n}\cdot\left(1-\frac{5}{n}\right)^2} =\infty. \]