ГДЗ по алгебре 11 класс Никольский Параграф 2. Предел функции и непрерывность Задание 18

\[\boxed{\mathbf{18.}}\]

\[\mathbf{Второй\ замечательный\ предел:}\]

\[\lim_{x \rightarrow \infty}(1 + x)^{\frac{1}{x}} = e.\]

\[\mathbf{а)\ }\lim_{x \rightarrow \infty}\left( 1 + \frac{1}{x} \right)^{3x}\]

\[x = \frac{1}{y}:\]

\[\lim_{x \rightarrow \infty}\left( (1 + y)^{\frac{1}{y}} \right)^{3} =\]

\[= \left( \lim_{x \rightarrow \infty}(1 + y)^{\frac{1}{y}} \right)^{3} = e^{3}.\]

\[\mathbf{б)\ }\lim_{x \rightarrow \infty}\left( 1 + \frac{1}{3x} \right)^{x}\]

\[x = \frac{1}{3y}:\]

\[\lim_{x \rightarrow \infty}(1 + y)^{\frac{1}{3y}} =\]

\[= \lim_{x \rightarrow \infty}\left( (1 + y)^{\frac{1}{y}} \right)^{\frac{1}{3}} =\]

\[= \left( \lim_{x \rightarrow \infty}(1 + y)^{\frac{1}{y}} \right)^{\frac{1}{3}} = e^{\frac{1}{3}}.\]

\[\textbf{в)}\ \lim_{x \rightarrow \infty}\left( 1 + \frac{1}{5x} \right)^{2x}\]

\[x = \frac{1}{5y}:\]

\[\lim_{x \rightarrow \infty}(1 + y)^{\frac{2}{5y}} =\]

\[= \lim_{x \rightarrow \infty}\left( (1 + y)^{\frac{1}{y}} \right)^{\frac{2}{5}} =\]

\[= \left( \lim_{x \rightarrow \infty}(1 + y)^{\frac{1}{y}} \right)^{\frac{2}{5}} = e^{\frac{2}{5}}.\]

\[\textbf{г)}\ \lim_{x \rightarrow \infty}\left( 1 - \frac{1}{4x} \right)^{2x}\]

\[x = - \frac{1}{4y}:\]

\[\lim_{x \rightarrow \infty}(1 + y)^{- \frac{2}{4y}} =\]

\[= \lim_{x \rightarrow \infty}\left( (1 + y)^{\frac{1}{y}} \right)^{- \frac{1}{2}} =\]

\[= \left( \lim_{x \rightarrow \infty}(1 + y)^{\frac{1}{y}} \right)^{- \frac{1}{2}} = e^{- \frac{1}{2}}.\]