2
$\begingroup$

Let $t\in (0,\pi)$ and $n$ change in natural numbers. I am wondering what is the answer to the following limit. $\lim_{n\rightarrow \infty} \frac{\sin(nt)}{\sin(t)}.$

Thank you.

  • 2
    Raymond and I interpreted the question differently in our answers. Since you started out with "Let $t\in(0,\pi)$", I thought that $t$ is fixed; but the question makes a lot more sense if it's intended to be about convergence of distributions as Raymond interpreted it. If so, I think you should clarify it.2012-07-18

2 Answers 2

6

Well $\ \frac {\sin(nt)}{\pi t}\to \delta(t)\ $ as $n\to\infty\ $ (equation (9)) so that (since $\frac t{\sin(t)}\to1$ as $t\to0$) : $\lim_{n\to \infty} \frac{\sin(nt)}{\sin(t)}=\pi \frac t{\sin(t)}\delta(t)=\pi\delta(t)$

  • 0
    I just want to make things clear.2012-07-21
9

$\sin t$ doesn't depend on $n$, so you can pull it out; so basically you're asking for the limit of $\sin(nt)$ for $n\to\infty$. Since $\sin x$ oscillates, there's no such thing.