Задача №2102

\[ \lim_{n\to\infty}\frac{(n+3)^3+(n+4)^3}{(n+3)^4-(n+4)^4}=\\ =\lim_{n\to\infty}\frac{(n+3)^3+(n+4)^3}{n^4+12n^3+54n^2+108n+81-\left(n^4+16n^3+96n^2+256n+256\right)} =\left[\frac{\infty}{\infty}\right]=\\ =\lim_{n\to\infty}\frac{(n+3)^3+(n+4)^3}{-4n^3-42n^2-148n-175} =\lim_{n\to\infty}\frac{\frac{(n+3)^3}{n^3}+\frac{(n+4)^3}{n^3}}{\frac{-4n^3-42n^2-148n-175}{n^3}}=\\ =\lim_{n\to\infty}\frac{\left(1+\frac{3}{n}\right)^3+\left(1+\frac{4}{n}\right)^3}{-4-\frac{42}{n}-\frac{148}{n^2}-\frac{175}{n^3}} =\frac{2}{-4} =-\frac{1}{2} \]