fag end

WordReference Random House Unabridged Dictionary of American English © 2025

fag′ end′, 

1. the last part or very end of something:
   the fag end of a rope.
2. Textiles the unfinished end of a piece of cloth;
   remnant.
• 1605–15

Collins Concise English Dictionary © HarperCollins Publishers::

fag end noun

1. the last and worst part, esp when it is of little use
2. the stub of a cigarette

Etymology: 17th Century: see fag²