As approaches 0, increases without bound. Thus in any open interval containing 0 there will be values of such that is a multiple of , values of such that is more than a multiple of , and values of such that is more than a multiple of . The sine of these values of will be 0, 1 and -1 respectively.
oscillates "an infinite number of times" between 1 and -1 in any neighbourhood of . Therefore the limit doesn't exist.
