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Thread: computing a limit

  1. #1
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    computing a limit

    Hi!

    The exercise is to compute the limit below if it exist. I know that I'm supposed to try to use the Monotone Conv. Thm or the Dominated Conv. Thm.

    $\displaystyle \lim\int_{0}^{\infty}\frac{\sin^n(x)}{x(1+x)}dx$


    My attempt:

    First i see that the MCT is not going to work here since $\displaystyle g_n\ge g_{n+1}$, where $\displaystyle g_n(x)=\frac{\sin^n(x)}{x(1+x)}$. So I guess I'm left with the DCT. But here I can't find an $\displaystyle g$ such that $\displaystyle g_n\rightarrow g$ a.e. (*) nor can I find an $\displaystyle f$ in $\displaystyle L^1$ such that $\displaystyle |g_n|\le f$. All I know for sure is $\displaystyle |g_n|\le 1/x(x+1)$ and this function is not in $\displaystyle L^1$.

    (*) actually I guess that $\displaystyle g=0$ since $\displaystyle \sin(x)<1$ except on a countable set.

    What to do?
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  2. #2
    Super Member girdav's Avatar
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    Re: computing a limit

    You can use the MCT for decreasing functions, since $\displaystyle g_1$ is integrable.
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