We consider discrete quasi-periodic long range operators with Liouvillean frequency. First, based on generalized Gordon type argument, we show that they can be approximated by a sequence of finite range operators which have no point spectrum for any phase. On the other hand, we show that when the potential for the dual model is small, then they can be approximated by a sequence of long range operators which have at least one eigenvalue for each phase in a set of full measure.