Задача №2105

\[ \lim_{n\to\infty}\frac{(n+6)^3-(n+1)^3}{(2n+3)^2+(n+4)^2} =\lim_{n\to\infty}\frac{n^3+18n^2+108n+216-\left(n^3+3n^2+3n+1\right)}{(2n+3)^2+(n+4)^2}=\\ =\lim_{n\to\infty}\frac{15n^2+105n+215}{(2n+3)^2+(n+4)^2} =\left[\frac{\infty}{\infty}\right] =\lim_{n\to\infty}\frac{\frac{15n^2+105n+215}{n^2}}{\frac{(2n+3)^2}{n^2}+\frac{(n+4)^2}{n^2}} =\lim_{n\to\infty}\frac{15+\frac{105}{n}+\frac{215}{n^2}}{\left(2+\frac{3}{n}\right)^2+\left(1+\frac{4}{n}\right)^2} =\frac{15}{5} =3. \]