\displaystyle
\lim_{x\rightarrow 0}(\sin xf(x))
\lim_{x\rightarrow 0}\left(\frac {\sin x} {x}xf(x)\right)=
\lim_{x\rightarrow 0}\frac {\sin x} {x}\lim_{x\rightarrow 0}xf(x)=
\lim_{x\rightarrow 0}xf(x).