McTag
When "there is a limited number of senses", we could also say that "the number of senses is limited".
"The number of sense are limited" doesn't sound right. "are" should be used if you are describing a quality of the senses themselves, not the number of them.
Like wandel said, the key word is "number", and regardless of the size of that number, it is never plural. Numbers are, however, but that's another story.