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Evaluate $\lim_{n \rightarrow \infty} \int_{[0,1]} x\cos(\frac{x}{n})$.

So, I'm pretty sure I'm supposed to use the Lebesgue dominated convergence theorem? I could use the Vitali convergence theorem, since the domain is finite. Struggling with this one, would appreciate some help, thanks!

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    $\begingroup$ the answer is $\frac{1}{2}$ right...? $\endgroup$ – Saketh Malyala Dec 8 '19 at 0:39
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    $\begingroup$ Hint: Use Lebesgue and that the integrand goes to $x$ as $n\to\infty$ $\endgroup$ – Maximilian Janisch Dec 8 '19 at 0:41
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$|x \cos(\frac x n) | \leq x$ and $x$ is integrable on $[0,1]$. Hence the limit is $\int_0^{1}x dx=\frac 1 2$.

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  • $\begingroup$ Okay, so we have a dominating function, but don't we also need the pointwise limit of $x cos(\frac{x}{n})$? Doesn't this function have no limit? $\endgroup$ – MSV Dec 8 '19 at 0:43
  • $\begingroup$ Wowww My mistake I see $\endgroup$ – MSV Dec 8 '19 at 0:44

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