Almost dead, one inspiring start of recovery: it’s on the line (9)
I believe the answer is:
numerator
'on the line' is the definition.
I can't judge whether this definition defines the answer.
'almost dead one inspiring start of recovery' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'dead' becomes 'numb' (I've seen this before).
'one inspiring' becomes 'erato' (muse or inspiration in Greek myth).
'start' means to remove the last letter (I've seen 'starts' mean this).
'of recovery' becomes 'or' (this might be a standard abbreviation of which I'm unaware).
'numb' with its final letter removed is 'num'.
'erato' with its final letter taken away is 'erat'.
'num'+'erat'+'or'='numerator'
'it's' is the link.
(Other definitions for numerator that I've seen before include "it goes over the line" , "the superior figure" , "Top half of 1/2 , for example" , "Upper part of fraction" , "Figure going over a line" .)