How to prove $\lim_{x\to0}\sin x\log x = 0$ How to prove $\lim_{x\to0}\sin x\log x = 0$

My try:

I tried using expansions of sin and log but that does not seem to work.
you can't use any taylor expansion for $\log(x)$

But you can say that $\sin x \sim x$, then you have $$\lim_{x \to 0} x\log x = 0$$

Because $x$ goes to $0$ so much faster than $\log x$ goes to infinity. 

Or if you have to justify that,  rewrite the limit as 

$$\frac{\log(x)}{\frac{1}{x}}$$ then use l'Hopital
lim x tends to 0 log sin x/1/x
=Limx tends to 0cosx/sinx/1/x^(-2)
=limx tends to 0 cotx/-1/x^2
=Limx tends to 0 -x^2/tanx
=Limx tends to 0 -2x/sec^2x
=0