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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. – user17762 Oct 24 '11 at 17:51
What did you try? Where did it go wrong? You will not learn without trying. – AD. Oct 24 '11 at 17:51
@AD. I don't have any clue. I was thinking about integration by parts type of techniques...TT – Alex J. Oct 24 '11 at 18:16
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? – Alex J. Oct 24 '11 at 18:37
three cases? oh when x=1! – Alex J. Oct 24 '11 at 19:02


$\arctan$ is bounded.

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