As the Wikipedia link says, the rhumb line reaches the pole in a finite distance. It is relatively intuitive. When you are moving along the rhumb line, you have to travel fixed distance for every degree of latitude (60 nautical miles if you're going due north or due south, 60/|cos(bearing)| nm for other bearings). Therefore, to go from the equator to the end of the line, you go 60*90/|cos(bearing)| nm.