The phrase "futile pursuit" means engaging in an activity or striving for something that is ultimately pointless or without any chance of success. Full definition
It may be quite entertaining to watch a small puppy set off in futile pursuit of the family cat. (thehappypuppysite.com)
Learning the law through cases can be a nearly futile pursuit. (obiter-dicta.ca)
It's impossible to reconcile the idea of a son alive with that of a son dead, so he shows the mother in bubbles of congealed time, invested in futile pursuits, as in the moment when, drowned in darkness, she calls a woman she does not know on the phone to ask her to be a character witness for her son. (sensesofcinema.com)