TITLE: limit $\lim_{n\to ∞}\sin(\pi(2+\sqrt3)^n)$
QUESTION [30 upvotes]: $\lim_{n\to ∞}\sin(\pi(2+\sqrt3)^n)$
I tried to write it as $\sin (n\pi - \theta)$ to get the form $∞-∞$ form within $\sin$ function. But could not proceed after that. How should I do it?
Edit:I am sorry, I forgot to mention $n\in \mathbb{N}$

REPLY [26 votes]: Hint(s): $(2+\sqrt{3})^n$ is closer and closer to an integer as $n$ increases, since
$$ (2+\sqrt{3})^n+(2-\sqrt{3})^n $$
is an integer and $|2-\sqrt{3}|<\frac{1}{3}$. The sine function is a Lipschitz-continuous function and $\sin(\pi m)=0$ for any integer $m$, hence your limit equals zero.

REPLY [10 votes]: since 
$$
K(n)=(2+\sqrt3)^n+(2-\sqrt3)^n \in \mathbf{N}
$$
$$\lim_{n\to ∞}(2-\sqrt3)^n=0$$
You can do the following:
$$\lim_{n\to ∞}\sin(\pi(2+\sqrt3)^n)=\lim_{n\to ∞}\sin(\pi(K(n)-(2-\sqrt3)^n))=-\lim_{n\to ∞}\sin(\pi(2-\sqrt3)^n)=0
$$