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Math Help - computing a limit

  1. #1
    Newbie
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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.

    \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 g_n\ge g_{n+1}, where 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 g such that g_n\rightarrow g a.e. (*) nor can I find an f in  L^1 such that |g_n|\le f. All I know for sure is |g_n|\le 1/x(x+1) and this function is not in L^1.

    (*) actually I guess that g=0 since \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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    Rouen, France
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    Re: computing a limit

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