certificate-lim-n-0-1-n-1-n-2-x-2-e-x-3-dx-pi-2-

Question Number 212600 by MrGaster last updated on 18/Oct/24

certificate :

\lim_{n \to \infty} \int_{0}^{1} \frac{n}{1 + n^{2} x^{2}} e^{x^{3}} d x = \frac{π}{2} .

Answered by Berbere last updated on 21/Oct/24

Extra \left or missing \right

= \frac{π}{2} e - \lim_{n \to \infty} \int_{0}^{1} 3 x^{2} e^{x^{3}} \tan^{- 1} (n x) d x

\frac{π}{2} - \frac{1}{x} ⩽ \tan^{- 1} (x) = \frac{π}{2} - \tan^{- 1} (\frac{1}{x}) ⩽ \frac{π}{2}

(\frac{π}{2} - \frac{1}{n x}) 3 x^{2} e^{x^{3}} ⩽ 3 x^{2} e^{x^{3}} \tan^{- 1} (n x) ⩽ \frac{π}{2} . 3 x^{2} e^{x^{3}}

\lim_{n \to \infty} \frac{π}{2} e - \frac{π}{2} - \frac{1}{n} \int_{0}^{1} 3 {x e}^{x^{3}} ⩽ u = \lim_{n \to \infty} \int_{0}^{1} 3 x^{2} e^{x^{3}} \tan^{- 1} (n x) ⩽ \frac{π}{2} e - \frac{π}{2}

\frac{π}{2} e - \frac{π}{2} - \frac{1}{n} \int_{0}^{1} 3 {x e}^{x^{2}} d x ⩽ \frac{π}{2} e - \frac{π}{2} - \frac{1}{n} \int_{0}^{1} 3 {x e}^{x^{3}} ⩽ u ⩽ \frac{π}{2} e - \frac{π}{2}

\lim_{n \to \infty} \frac{π}{2} e - \frac{π}{2} - \frac{3}{2 n} (e - 1) ⩽ u ⩽ \frac{π}{2} e - \frac{π}{2}

u = \frac{π}{2} e - \frac{π}{2}; A = \frac{π}{2} e - u = \frac{π}{2}

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