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 Post subject: Limit of an integral
PostPosted: Sun Aug 06, 2017 6:43 am 
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The following exercise is just an alternative of IMC 2017/2/1 problem. It is quite easy but it's not a bad idea to have it here as well.

Given the continuous function $f:[0, +\infty) \rightarrow \mathbb{R}$ such that $\lim \limits_{x \rightarrow +\infty} x^2 f(x) = 1$ prove that

$$\lim_{n \rightarrow +\infty} \int_0^1 f(n x)\, {\rm d}x =0$$

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 Post subject: Re: Limit of an integral
PostPosted: Sat Sep 09, 2017 8:17 am 

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Since $\lim \limits_{x \rightarrow +\infty} f(x) =0$ the result follows immediately by making the change of variables $u=nx$ . :)

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