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Putnam 2011 A3

Source:

December 5, 2011
Putnamlimitintegrationtrigonometrycalculusratioinequalities

Problem Statement

Find a real number cc and a positive number LL for which lim⁡r→∞rc∫0π/2xrsin⁡x dx∫0π/2xrcos⁡x dx=L.\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.
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