# Solution to $\lim_{n\rightarrow\infty}\int_{-\infty}^\infty\frac{\sin^{\circ n}x}{x}dx$

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 $$-5$$.

• I'm working on a proof now, but I believe the answer is that it approaches zero because the numerator of your expression approaches zero uniformly, and therefore the entire expression should approach zero uniformly everywhere except for an arbitrarily small interval around 0. Apr 5, 2023 at 23:34
• This may or may not help: math.stackexchange.com/q/4131716
– Gary
Apr 6, 2023 at 0:01
• A little hint: $\forall \epsilon > 0$, $\exists N \in \mathbb{N}$ such that for $n > N$, we get $|\sin(\sin(\sin(\ldots(x)\ldots))| < \epsilon$ Apr 6, 2023 at 3:28

## 1 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 $$0$$.

• Not sure why this is downvoted. Anyone care to comment? Apr 13, 2023 at 19:03
• @kevinkayaks I think this problem of random downvotes must be taken care of. If only the downvoter can get recorded... Apr 13, 2023 at 19:04
• I really don't understand this answer. What does the equation $a_x=\sin(a_x)$ have to do with the integral? Apr 13, 2023 at 19:18
• @KamalSaleh: Granted, $f_n(x):=\frac{\sin^{n\circ}(x)}{x}\xrightarrow{n\rightarrow\infty}0$. Buț why does that imply that $\int f_n(x)\,dx\xrightarrow{n\rightarrow\infty}0$? Dominated convergence does not apply here in a straightforward way. some additional analysis is required. Apr 13, 2023 at 20:00