Monday, 3 June 2019

calculus - Prove the existence of a limit : limlimitsxrightarrow+inftyint+inftyvarepsilonxF(xt)costdt=0

Let F(x),G(x) be nonnegative decreasing functions in [0,+), withlim



(1) Prove that: \forall \varepsilon>0,we have \displaystyle \lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}}=0
(2) If we have \lim_{n\rightarrow+\infty}{\int_{0}^{+\infty}{(F(t)-G(t))\cos{\frac{t}{n}} dt}}=0
then prove that
\lim_{x\rightarrow0}{\int_{0}^{+\infty}{(F(t)-G(t))\cos{(xt)}dt} }=0



I tried let f(x)=\lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}} ,then for a fixed value of x,by Dirichlet test,we can see the f(x)=\lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}}

is convergence,then I have no idea about the next step.:(

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