# Math Help - computing a limit

1. ## 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?

2. ## Re: computing a limit

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