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let-U-n-0-cos-nsinx-sin-ncosx-x-2-3-2-dx-determine-lim-n-U-n-and-lim-n-e-n-2-U-n-




Question Number 137696 by Mathspace last updated on 05/Apr/21
let U_n =∫_0 ^∞  ((cos(nsinx)−sin(ncosx))/((x^2  +3)^2 ))dx  determine lim_(n→+∞) U_n   and lim_(n→+∞) e^(−n^2 )  U_n
$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({nsinx}\right)−{sin}\left({ncosx}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$$${determine}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} {e}^{−{n}^{\mathrm{2}} } \:{U}_{{n}} \\ $$

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