245 Give it one's best shot

(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.