(ITA-1952) Calcular LIMITE III

por **Jigsaw** Qua 18 Out 2023, 19:20

3.2) Calcular o [latex]\lim_{n \rightarrow \infty}\frac{\sqrt[n]{n!}}{n}[/latex].

Spoiler:

por **Giovana Martins** Qua 18 Out 2023, 21:36

[latex]\\\mathrm{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L=\lim_{n\to \infty }\frac{\sqrt[\mathrm{n}]{\mathrm{n!}}}{n} =\lim_{n\to \infty }\sqrt[\mathrm{n}]{\frac{\mathrm{n!}}{\mathrm{n^n}}} =\lim_{n\to \infty }\sqrt[\mathrm{n}]{\mathrm{\frac{n\cdot ...\cdot 3.2.1}{n\cdot ...\cdot n\cdot n}}} }\\\\ \mathrm{ln(L)=\lim_{n\to \infty }ln\left ( \mathrm{\frac{n\cdot ...\cdot 3.2.1}{n\cdot ...\cdot n\cdot n}} \right )^{-n}=\lim_{n\to \infty}\frac{1}{n}\left [ ln\left ( \frac{1}{n} \right )+ln\left ( \frac{2}{n} \right )+...+ln\left ( \frac{n}{n} \right ) \right ]}\\\\ \mathrm{\ \ \ ln(L)=\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^{n}\left [ ln\left (\frac{i}{n} \right )\right ]=\int_{0}^{1}ln(x)dx=-1\ \therefore\ L=\lim_{n\to \infty }\frac{\sqrt[\mathrm{n}]{\mathrm{n!}}}{n} =\frac{1}{e}}[/latex]

(ITA-1952) Calcular LIMITE III

(ITA-1952) Calcular LIMITE III

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