What is the limit of $n\sin \left(\frac{1}{n}\right)$ as $n \to \infty$ The textbook I'm reading gives no proof of the limit being 1 apart from a flimsy 0/0, and has covered only the "epsilon" and sandwich test for co...--prophetes.ai

What is the limit of $n\sin \left(\frac{1}{n}\right)$ as $n \to \infty$ The textbook I'm reading gives no proof of the limit being 1 apart from a flimsy 0/0, and has covered only the "epsilon" and sandwich test for convergence to a particular limit. Do not rearrange to turn $n$ into $\frac{1}{n}$, that is the source of the flimsy $0/0$ argument.

Hint: $$\lim_{x\to 0}\frac{\sin x}x=1.$$