Задача №2115

\[ \lim_{n\to\infty}\frac{(n+1)^3-(n-1)^3}{(n+1)^2+(n-1)^2} =\lim_{n\to\infty}\frac{n^3+3n^2+3n+1-\left(n^3-3n^2+3n-1\right)}{n^2+2n+1+n^2-2n+1}=\\ =\lim_{n\to\infty}\frac{6n^2+2}{2n^2+2} =\lim_{n\to\infty}\frac{3n^2+1}{n^2+1} =\left[\frac{\infty}{\infty}\right] =\lim_{n\to\infty}\frac{\frac{3n^2+1}{n^2}}{\frac{n^2+1}{n^2}} =\lim_{n\to\infty}\frac{3+\frac{1}{n^2}}{1+\frac{1}{n^2}} =\frac{3}{1} =3 \]