Monday, 28 January 2019

real analysis - Calculate: limntoinftyintpi/20frac11+xtannxdx



I'm supposed to work out the following limit:




lim



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


Answer



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}.



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