Задача №2107

\[ \lim_{n\to\infty}\frac{(n+10)^2+(3n+1)^2}{(n+6)^3-(n+1)^3} =\lim_{n\to\infty}\frac{(n+10)^2+(3n+1)^2}{n^3+18n^2+108n+216-\left(n^3+3n^2+3n+1\right)}=\\ =\lim_{n\to\infty}\frac{(n+10)^2+(3n+1)^2}{15n^2+105n+215} =\left[\frac{\infty}{\infty}\right] =\lim_{n\to\infty}\frac{\frac{(n+10)^2}{n^2}+\frac{(3n+1)^2}{n^2}}{\frac{15n^2+105n+215}{n^2}} =\lim_{n\to\infty}\frac{\left(1+\frac{10}{n}\right)^2+\left(3+\frac{1}{n}\right)^2}{15+\frac{105}{n}+\frac{215}{n^2}} =\frac{10}{15} =\frac{2}{3}. \]