[tex]x\sin \frac{1}{x}\overset{x\rightarrow 0}{\longrightarrow }0\sim \forall \varepsilon > 0 \; \exists \delta (\varepsilon ) > 0:\forall x \; 0 < |x| < \delta \rightarrow \left | x\sin \frac{1}{x} \right | < \varepsilon \\\delta (\varepsilon )=\varepsilon \Rightarrow \left | x\sin \frac{1}{x} \right |\leq |x| < \delta =\varepsilon[/tex]