(PW) try very hard
I gave it my best shot, but I still didn’t make the team.
(Wiki) to give it one's best shot
(idiomatic) To make one's best effort or attempt; to try as hard as possible.
I didn't win the contest, but I sure gave it my best shot.
(dictionary)
If you give something your best shot, you do it as well as you possibly can.
INFORMAL
give something your best shot phrase V inflects
I don't expect to win. But I am going to give it my best shot.
