Задача №1081

\[ \lim_{x\to\infty}\frac{(x+1)^{10}+(x+2)^{10}+\ldots+(x+100)^{10}}{x^{10}+10^{10}} =\left[\frac{\infty}{\infty}\right] =\lim_{x\to\infty}\frac{\frac{(x+1)^{10}+(x+2)^{10}+\ldots+(x+100)^{10}}{x^{10}}}{\frac{x^{10}+10^{10}}{x^{10}}}=\\ =\lim_{x\to\infty}\frac{\left(1+\frac{1}{x}\right)^{10}+\left(1+\frac{2}{x}\right)^{10}+\ldots+\left(1+\frac{100}{x}\right)^{10}}{1+\frac{10^{10}}{x^{10}}} =\frac{100}{1} =100. \]