Prove $\lim _{x \to 0} \sin(\frac{1}{x}) \ne 0$ Prove $$\lim _{x \to 0} \sin\left(\frac{1}{x}\right) \ne 0.$$
I am unsure of how to prove this problem. I will ask questions if I have doubt on the proof. Thank you!

in order to prove the limit $\not=0$, only need to find a sequent $\{x_n\}\rightarrow0$,but $\sin(x_n) \not\rightarrow 0$. and the below is the construction.
$$x_n=\frac{1}{\frac{\pi}{2}+2n\pi}$$
$$\sin\left(\frac{1}{x_n}\right)=1$$
