Question: Consider: $$2x\sin(1/x) - \cos(1/x)$$.

How to show $f(x)$ is bounded?

Thanks!

Answer: 
set $x=1/t$ then the expression becomes:
$$\frac{\sin(t)}{t} - \cos(t)$$.

It is clearly bounded since

$$\left|\frac{\sin(t)}{t}\right| \le 1, \left|\cos(t)\right| \le 1$$
**Hint:** Note that

$$\left|2x\sin \frac1x-\cos \frac1x\right| \leq 2\left|\frac{\sin\frac1x}{\frac1x}\right| +\left|\cos\frac1x\right|$$

Can you see that the terms in vertical brackets are each bounded by $1$? What then?