Is sin(1/x) periodic?with what period time? Is sin(1/x) periodic? with what period time?

No, it's not. If it were periodic, the derivative would be periodic too (I'll let you prove this result). But here,
$$f'(x)=\frac{-\cos\left(\frac{1}{x}\right)}{x^2}\to 0 \ \text{ as } \ x \to\infty, $$
and so, it's impossible that $x\mapsto \sin\left(\frac{1}{x}\right)$ is periodic.
