How do I show $$\lim_{n \to \infty} \int_0^\infty \frac{n}{n^2+x}\sin\left(\frac{1}{x}\right)\, dx = 0\,\,?$$ I've tried splitting into the cases where $x \leq 1$ and $x \geq 1$ but I am having trouble finding bounds so that I can apply the dominated convergence theorem.
How to show $\lim_{n \to \infty} \int_0^\infty \frac{n}{n^2+x}\sin(\frac{1}{x})\, dx = 0\,$
limitsmeasure-theoryreal-analysis
Best Answer
Edit: The second half of this is nonsense. See the comments below...
Say the integrand is $f$. If $0<x\le1$ then $|f(x)|\le 1$, while if $x\ge1$ then $|f(x)|\le 1/x^2$, since $|\sin(t)|\le|t|$.