This was a question that I thought about as I was playing around with the cardinal sine function:
Find$$\lim_{n\rightarrow\infty}\int_{-\infty}^\infty\frac{\sin^{\circ n}x}{x}dx$$ with ${}^\circ$ denoting functional composition.
I don't really know how to approach this question, but intuition tells me that this approaches $\pi$. Maybe it doesn't even converge. I have no idea.
WA says that for $n=3$ we get something around $-5$.
Best Answer
Credit goes to @Gary's comment:
We will study the integrals. Let $$\sin(\sin(\sin...(x)...))=a_x$$Then$$a_x=\sin(a_x)$$Or $a_x=0$. This means that the function in the integrand is really just $0$ and the answer is $0$.