To do your best is to try as hard as you possibly can. Nothing left in reserve, whatever the result turns out to be.
The possessive changes with the person — I did my best, they did theirs — which is what makes it a pattern rather than a fixed string. It is the standard thing said to a child before an exam and the standard thing said afterwards, and both times it means the effort counts and the mark does not. Best of luck is the neighbour, and try your best is the same promise made in advance.
1phrasedo your bestTo try as hard as you possibly can.
“Just do your best and forget the rest of them.”
do your best is a full effort.
Practised in Common Word Pairs