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How would I evaluate these limits? $$ \lim_{n \to \infty} \int_0^\infty \frac{n}{1+(nx)^2} \ dx$$ and $$ \lim_{n \to \infty} \int_0^\infty \frac{(1+(nx)^2)}{(1+nx^2)^n} \ dx$$

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    @jessica: Where are they from? What theorems do you know? What do you know about the pointwise limit of the integrands?2011-02-05
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    @PEV: Thanks for editing. Even i was looking to edit it.2011-02-05
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    @jessica: What convergence theorems are you aware of? For the first one at least, you could evaluate it directly.2011-02-05
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    right now im learning fatou's lemma, mct, and dct. the integrand of the first question goes to 0. i was thinking of finding a dominating function, ie 1/nx^2. im not sure how to proceed from here.2011-02-05
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    The first integral is trivial; Doing a change of variable $u=nx$ you get the integral of $\frac{1}{1+u^2}$, so $$\int_0^{\infty}\frac{n}{1+(nx)^2}\,dx = \lim_{a\to\infty}\int_0^{na}\frac{du}{1+u^2} = \lim_{a\to\infty}\arctan(na) = \frac{\pi}{2}$$ so the sequence is constant.2011-02-05
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    oh i see! its so obvious now, it shouldve been a calc 1 problem, i dont even need the convergence theorems2011-02-05
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    any tips on the 2nd ?2011-02-06
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    Did you get something out of one of the answers below?2011-04-07

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