12
$\begingroup$

Find with proof the following limit:

$\lim_{n \to \infty} \int_{-\infty}^{\infty} \frac{(\sin(x))^n}{x^2}dx$

I want to use the DCT but I cannot seem to dominate $f_{n}(x)=\frac{(\sin(x))^n}{x^2}$ by an integrable function. Any help would be appreciated.

1 Answers 1

14

Put $f_n(x):=\frac{(\sin x)^n}{x^2}\mathbf 1_{x\neq 0}$. We have for $n\geq 2$ and $x\in\mathbb R$ \begin{align*}|f_n(x)|&=\frac{|(\sin x)^n|}{x^2}\mathbf 1_{|x|\geq 1} +\frac{|(\sin x)^n|}{x^2}\mathbf 1_{|x|< 1}\mathbf 1_{x\neq 0}\\ &\leq \frac 1{x^2}\mathbf 1_{|x|\geq 1}+x^{n-2}\mathbf 1_{|x|< 1}\mathbf 1_{x\neq 0}\\ &\leq\frac 1{x^2}\mathbf 1_{|x|\geq 1}+\mathbf 1_{|x|< 1}=:g(x) \end{align*} which is an integrable function. Since $f_n(x)\to 0$ if x\notinĀ \frac{\pi}2+\pi \mathbb Z, a set of measure $0$, we can apply the dominated convergence theorem to get $\lim_{n\to\infty}\int_{-\infty}^{+\infty}\frac{(\sin x)^n}{x^2}dx=0.$