lim-x-x-x-x-1-x-2-4-1-3-4-6-3-5-2-1-2-3-4-3-4-5-6-5-6-7-pi-2-lim-x-x-x-x-1-x-2-1-4-3-4-3-6-5-1-e-pi-2-1-e

Question Number 118967 by obaidullah last updated on 21/Oct/20

\lim_{x \to \infty} {({(\frac{x!}{x^{x}})}^{\frac{1}{x}})}^{\frac{2 \cdot 4}{1 \cdot 3} \cdot \frac{4 \cdot 6}{3 \cdot 5} \cdot \cdot \cdot} = ?

\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \cdot \cdot = \frac{π}{2}

\lim_{x \to \infty} {({(\frac{x!}{x^{x}})}^{\frac{1}{x}})}^{\frac{2}{1} \cdot \frac{4}{3} \cdot \frac{4}{3} \cdot \frac{6}{5} \cdot \cdot \cdot} = {(\frac{1}{e})}^{\frac{π}{2}} = \sqrt{{(\frac{1}{e})}^{π}} = \frac{1}{\sqrt{e^{π}}}