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I'm supposed to work out the following limit:

$\lim_{n\to\infty} \int_{0}^{\pi/2}\frac{1}{1+x \left( \tan x \right)^{n} }dx$

I'm searching for some resonable solutions. Any hint, suggestion is very welcome. Thanks.

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    @lvb Or even the monotone convergence theorem, $I$ guess. At least $f$or one of the two pieces.2012-06-22

1 Answers 1

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Note that the integrand is bounded in $[0,\pi/2]$, so if $\lim_{n\to \infty} \frac{1}{1+x\tan^nx}$ exists a.e. then we may apply the Dominated Convergence Theorem to show $\lim_{n\to \infty} \int_0^{\pi \over 2}\frac{1}{1+x\tan^nx}dx = \int_0^{\pi \over 2}\lim_{n\to \infty} \frac{1}{1+x\tan^nx}dx.$

If $x<\pi/4$ then the integrand converges to 1, and if $x>\pi/4$ then it converges to 0. Thus we have the integral equals $ \int_0^{\pi \over 4} 1dx + \int_{\pi \over 4}^{\pi \over 2} 0dx = \frac{\pi}{4}. $