Note: This is for a Calculus 1 course that has just started, so appropriately I can only use algebra/trig to solve these

$\displaystyle \lim_{x \to \infty } x \sin \frac{1}{x}$

I know sin(1/x) is always between 0 and 1, but sin(1/x) is getting close to 0 as x approaches infinity (sin of something close to 0 is close to 0), and so I'm getting infinity * 0 which doesn't really make any sense.

How can I show my work algebraically?