Answer
Limit does not exist.
Work Step by Step
Let us consider that $\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\sin (\dfrac{n \pi}{6})$
When $n$ is odd, then we have: $\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\sin (\dfrac{n \pi}{6})=\pm 1, \pm \dfrac{1}{2}$
When $n$ is even, then we have: $\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\sin (\dfrac{n \pi}{6})=\dfrac{\pm \sqrt 3}{2}$
We do not get same value of the limit. $nx$. So, the limit does not exist.
