Find $\lim_{x \to 0} x \cdot \sin{\frac{1}{x}}\cos{\frac{1}{x}}$ $$\lim_{x \to 0} x \cdot \sin{\frac{1}{x}}\cos{\frac{1}{x}}$$
I don't solve this kind of limits, I can't try anything because it seems difficult to me.

Let $\epsilon>0$ be given. Now consider $ |x\sin \frac{1}{x}\cos\frac{1}{x}|\leq|x|<\delta=\epsilon$, since sin and cos functions are bounded. So, the required limit is equal to 0.
