I should prove that there is no limit at $ x_0 = 0 $ in the function $$ f(x) = \sin({\frac{1}{x}}) $$with the sequences $ x_n = 1/(n\pi) $ and $ x_n= 1/(2n\pi + \pi/2) $

What would be the approach to do this?

My 2 sequences converging to zero as $ n \rightarrow \infty$

Is there any theorem?

## Best Answer

You wanna show that you have two subsequencies converging to 0 while the function series are not consistent. That is $$x_n=\frac{1}{n\pi}\rightarrow 0$$ $$y_n=\frac{1}{2n\pi+\pi/2}\rightarrow 0$$ while $$\sin\frac{1}{x_n}=\sin(n\pi)\rightarrow 0$$ $$\sin\frac{1}{y_n}=\sin(2n\pi+\pi/2)\rightarrow 1$$