In general, whene you have a clogged function (like \sin or \cos) times 0 the \lim will be 0. It is usefull to think like this! Taking the inequalities and sandwich theorem will allways help!
I usually face these limits like this: |x \sin(\frac 1 x) | \leq |x| ⇔ -|x| \leq x \sin(\frac 1 x)...