1. I’m going to try a last-ditch effort to uproot this old oak tree myself before I call the tree company to come and do it by machine.
2. Ronnie slipped and fell as he ran to catch the baseball, but when he looked up, the ball was still sailing through the air. Ronnie got up and made a last-ditch effort to catch the ball.

The expression often conveys a sense of great physical effort and is usually used when the outcome is likely to be unsuccessful.