An intuitive method for seeing the answer is to consider the extreme cases.
Say that . This implies that the , which means that and . So then it must be the case that . Then you know that . As a result, knowing that gives you information about , specifically that must be zero.
Now say that . That implies that . That means that you could have (in the case where ), or you could have (in the case where ) or anything in between. Less information is known about the value of now, when takes a larger value.
When , you are sure that but when , could be anything from to . So the smaller that is, the more certain you are about the values that can take. By definition, then, because the value of depends on the value of , they cannot be independent.