The Collaborative International Dictionary of English v.0.48:

Fag-end \Fag"-end"\, n.
   1. An end of poorer quality, or in a spoiled condition, as
      the coarser end of a web of cloth, the untwisted end of a
      rope, etc.
      [1913 Webster]

   2. The refuse or meaner part of anything.
      [1913 Webster]

            The fag-end of business.              --Collier.
      [1913 Webster]