give it (one's) best shot
To put forth one's best effort in an attempt to do or accomplish something. Henry gave it his best shot, but the board still denied his proposal.
give it one's best shot
Make one's hardest or most enthusiastic attempt, as in I don't know if I can do it, but I'll give it my best shot. This expression employs shot in the sense of "attempt," a usage dating from the mid-1700s.
give it your best shot
try as hard as you can to do something. informalgive it your best ˈshot
try as hard as you can to do or achieve something: I probably won’t win the game, but I’ll give it my best shot.