If you are greatly concerned about missing your connection, it might be worth considering going standby for an earlier LAX-DEN flight. Confirming a seat on an earlier flight that same day is possible for a $75 fee (waived for United 1K's), starting up to 3 hours prior to the flight you'd like to be on. Standing by and taking your chances at the gate without confirmation is free.

Some useful info here: Everything you wanted to know about the $75 confirmed same-day change process