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I have this limit to evaluate $$\lim\nolimits_{n \rightarrow +\infty} \int_{0}^{2} \arctan \left(\frac{1}{1+x^n}\right) dx.$$

I have no idea how to solve this homework problem. Help!

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    HINT: Can you find a function which dominates $\arctan \left(\frac1{1+x^n} \right)$?. Find $\lim_{n \rightarrow \infty} \arctan \left(\frac1{1+x^n} \right)$ and then use Lebesgue dominated convergence theorem to swap the limit and the integral.2011-10-24
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    What did you try? Where did it go wrong? You will not learn without trying.2011-10-24
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    @AD. I don't have any clue. I was thinking about integration by parts type of techniques...TT2011-10-24
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    No, integration by parts is not the way. Try the hint already given in a comment.2011-10-24
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    So for x between 0 and 1, the $\lim_{n ->\infty} \arctan(\frac{1}{1+x^n})$ is $\arctan 1$, which is $\pi/4$. for x between 1 and 2, it's $\arctan 0$, which is 0?2011-10-24
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    If I should be picky you there are three cases/limits...2011-10-24
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    three cases? oh when x=1!2011-10-24
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    @TaoleZ Right! But it does not matter in the situation :)2011-10-24
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    When you think about Lebesgue integrals, think about convergence theorems.2011-10-24
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    Anytime you have a limit and a Lebesgue integral your mind should involuntarily think "this requires a convergence theorem!"2011-10-24

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