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do (one's) best
To do as well as one possibly can at something. Primarily heard in UK, Australia. I'm just not good at math, so believe me, a B- in Algebra means that I've done my best. No, you're not the star player on the team, but you always do your best, which encourages the rest of us to do the same.
do one's (level) best
to do something as well as one can. Just do your level best. That's all we can ask of you. Tom isn't doing his best. We may have to replace him.