2: You should get
Note that the moment there appeared
Anyway, the limit expression you face now is an indeterminate product, so apply the standard procedure. Change the product into a ratio by putting something "under", probably the sine, since the logarithm would be more complicated; another reason is that leaving the sine on top would not improve it (derivative changes it into a cosine that behaves similarly), while leaving logarithm up will allow the derivative to get rid of it.