$\large f(x) = \frac{1}{ 2+ \lfloor \sin x\rfloor}$

If $a_1$ is the largest value of $f(x)$ above and $a_{n+1} = \dfrac{ (-1)^{n+2} }{n+1} + a_{n}$, for $n \ge 1$, find the value of $\displaystyle \lim_{n \to \infty} a_{n}$.

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.