Задача №2109

\[ \lim_{n\to\infty}\frac{(n+7)^3-(n+2)^3}{(3n+2)^2+(4n+1)^2} =\lim_{n\to\infty}\frac{n^3+21n^2+147n+343-\left(n^3+6n^2+12n+8\right)}{(3n+2)^2+(4n+1)^2}=\\ =\lim_{n\to\infty}\frac{15n^2+135n+335}{(3n+2)^2+(4n+1)^2} =\left[\frac{\infty}{\infty}\right] =\lim_{n\to\infty}\frac{\frac{15n^2+135n+335}{n^2}}{\frac{(3n+2)^2}{n^2}+\frac{(4n+1)^2}{n^2}} =\lim_{n\to\infty}\frac{15+\frac{135}{n}+\frac{335}{n^2}}{\left(3+\frac{2}{n}\right)^2+\left(4+\frac{1}{n}\right)^2} =\frac{15}{25} =\frac{3}{5}. \]