do (one's) best

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do (one's) best

To do as well as one possibly can at something. 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.
Farlex Dictionary of Idioms. © 2015 Farlex, Inc, all rights reserved.

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.
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs. © 2002 by The McGraw-Hill Companies, Inc.

do one's best

Also, do one's level best or one's damnedest . Perform as well as one can, do the utmost possible, as in I'm doing my best to balance this statement, or She did her level best to pass the course, or He did his damnedest to get done in time. The first term dates from the 16th century, but the addition of level, here meaning "very," occurred only in the mid-1800s; the variant dates from the late 1800s.
The American Heritage® Dictionary of Idioms by Christine Ammer. Copyright © 2003, 1997 by The Christine Ammer 1992 Trust. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.
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References in classic literature ?
We have certainly done our best; and most fortunately having it in our power to introduce you to very superior society, and, from our connection with Rosings, the frequent means of varying the humble home scene, I think we may flatter ourselves that your Hunsford visit cannot have been entirely irksome.
"We've done our best, sir," the station-master remarked with a note of self-congratulation in his tone.
She said: "We've done our best to try and make sure everybody knows what's happening but I still think there are parents that don't know about it."
We are riparian land owners and whilst we have done our best to keep our part of the river clear flowing we can do nothing about what is happening downstream of us.
Do as you like with it, they seemed to say, We've done our best to show you the world and can do no more.
On the following page, we have done our best to identify facilities throughout the United States that produce 20,000 metric tons or more per year of secondary aluminum or otherwise consume significant amounts of aluminum scrap.
But we've done our best to keep our key people in place-- and even counterintuitively we have hired people in this buyer's market.
Over the years, we have done our best to alert readers--and the general public--to the astronomical levels of calories, fat, sodium, and sugar that some of those foods supply.
We've done our best, we've trained well, and we're here to give Spain a good fight.
Asked what his greatest management challenge was this past year, he said, "We've done our best to keep our key people in place, and even counterintuitively we have hired people in this buyer's market."