Computing limit of $\frac{1}{n \sin \left(\pi ^6 n\right)}$ over integers

Question

I would like to compute the foillowing limit: $$\lim_{n \to \infty}\; \frac{1}{n \sin(\pi^6 n)}$$ assuming that $n$ is an integer number.

I use this code, but it does not give an answer.

Limit[1/(a Sin[a π^6]), a -> ∞, 
 Assumptions -> a > 0 && a ∈ Integers]

Answer 1

An appropriate tool for calculating limits of sequences over integers is DiscreteLimit. However DiscreteLimit[1/(a Sin[π^6 a]), a -> ∞] cannot compute our task. On the other hand we can figure out that DiscreteLimit[a Sin[π^6 a], a -> ∞] yields Indeterminate, i.e. it says that the limit does not exist and. We can also find it calculating discrete limes superior and limes inferior:

DiscreteMaxLimit[ a Sin[π^6 a], a -> ∞]
DiscreteMinLimit[ a Sin[π^6 a], a -> ∞]

 ∞
-∞

Alternatively one can find it with standard limes superior and limes inferior, e.g. Through @ { MinLimit, MaxLimit}[1/(a Sin[π^6 a]), a -> ∞]. For an insight it is reasonable to plot appropriate sequence

DiscretePlot[ 1/(a Sin[π^6 a]), {a, 1000, 1240, 2}, ImageSize -> Large]

It is clearly seen that $\sin( \pi a)$ takes values between $-1$ and $1$, however $\sin(\pi^6 a)$ never equals but it can approach $0$ with a very good approximation for appropriately large integer values of $a$. E.g. we find $6$ values of $\sin(\pi^6 a)$ in the first $10^6$ natural numbers $a$ closest to $0$:

N[ TakeSmallestBy[ Sin[π^6 Range[10^6]], Abs, 6], 10]

 {-1.694781536*^-6, 3.389563072*^-6, -5.084344608*^-6, 6.779126144*^-6, 
  -8.47390768*^-6, 0.00001016868922}

and for our sequence they are

1/%

{-590046.5510, 295023.2755, -196682.1837, 147511.6377,
 -118009.3102, 98341.09183}

Answer 2

 DiscreteLimit[(a Sin[Pi^6 a]), a -> Infinity]
                (*Indeterminate *)
    DiscretePlot[(a Sin[Pi^6 a]), {a, 1, 1000, 1}]

so $\lim_{n \to \infty}\; \frac{1}{n \sin(\pi^6 n)}$ divergence.

Or a method using series expansion

fx = Series[1/((1/x)* Sin[Pi^6 *(1/x)]), {x, 0, 1}] // Normal(* find the sereis expansion of it *)
Plot[fx, {x, -1, 1}]
Limit[fx, x -> 0]