Find the limit as n approaches infinite

We have the following function:

$$U_n = \sin \dfrac{1}{3} n \pi$$

What is the limit of this function as n approaches infinity?

I first tried to use my calculator as help, for n I chose some arbitrary large numbers, such as 100 and 1000. Then I I just took $n = 10^{50}$ and it gave me an error.

So the correct answer is it doesn't have one, but why? Why does this function have a solution for $n = 10^2, 10^3$ and not for bigger numbers such as $n=10^{50}$?

Answer

As David hinted in his comment, try using the fact that

$$\sin(x + 2\pi k) = \sin(x)$$
whenever $k$ is an integer. Or, if you just write out the first few values of $U_n$ (compute these using the unit circle) and you should notice a pattern.