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

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

 ∞
-∞

For an insight it is reasonable to plot appropriate sequence

DiscretePlot[ 1/(a Sin[Pi^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 $-1$, $0$ and $1$ with a very good approximation for appropriately large integer values of $a$.

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}

Appropriate tool for calculating limits of sequences over integers is DiscreteLimit. Nevertheless it cannot compute our task. The limit does not exist. We can find it with calculating limes superior and limes inferior:

MaxLimit[1/(a Sin[Pi^6 a]), a -> ∞]
MinLimit[1/(a Sin[Pi^6 a]), a -> ∞]

 ∞
-∞

For an insight let's plot appropriate sequence

DiscretePlot[1/(a Sin[Pi^6 a]), {a, 1000, 1200}]

It is clearly seen that Sin[ Pi a] takes values between -1 and 1, however Sin[Pi^6 a] can approach -1, 0 and 1 with a very good approximation.

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

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

 ∞
-∞

For an insight it is reasonable to plot appropriate sequence

DiscretePlot[ 1/(a Sin[Pi^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 $-1$, $0$ and $1$ with a very good approximation for appropriately large integer values of $a$.