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