Result of $\lim_{n\to\infty} \cos(\pi n)\sin(\pi n)$

Question

Task is to solve $\lim_{n\to\infty} \cos(\pi n)\sin(\pi n)$ for $n\in\Bbb Z$ (not $n\in\Bbb R$). Is there any way to find definite solution?

Thank you for your help.

Answer 1

Recall that

$$\sin (2x)=2\sin x \cos x$$

then we have that the expression

$$\cos(\pi n)\sin(\pi n)=\frac12 \sin(2\pi n)=\frac12 \sin (2\pi)=0$$

is identically zero and thus

$$\lim_{n\to\infty} \cos(\pi n)\sin(\pi n)=0$$