From Longman Dictionary of Contemporary Englishgive something your best shotgive something your best shotto make as much effort as you can to achieve something difficult This case is going to be tough, but I promise I’ll give it my best shot. Lydia didn’t get the job, but at least she gave it her best shot. → shot
Examples from the Corpus
give something your best shot• Hopefully he can recover and regain his test place and give it his best shot.• The band gave it their best shot, until the arrival of the blue meanies put an end to the proceedings.• I'd have given it my best shot, and that was all anyone could demand from me.• I just have a feeling that we have given it our best shot.• I'm not promising I'll succeed, but I'll give it my best shot.• You were never entirely safe from prying fingers in Chinatown, but I had to give it my best shot. 