Is $\sin\left(\frac{1}{x}\right)$ periodic? with what period time?
2026-03-29 20:16:42.1774815402
Is $\sin\left(\frac{1}{x}\right)$ periodic?with what period time?
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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.