Application of Dominated Convergence Theorem?

Solution 1:

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 $x\notin \frac{\pi}2+\pi \mathbb Z$, a set of measure $0$, we can apply the dominated convergence theorem to get $$\lim_{n\to\infty}\int_{-\infty}^{+\infty}\frac{(\sin x)^n}{x^2}dx=0.$$