The Liberty Stadium side fought back bravely in Le Mans but a late penalty condemned them to an 18-12 defeat.
Their hopes of qualifying for the last eight of Europe’s premier competition ended with the French defeat, but Biggar hailed his side’s battling qualities.
He said: “If you don’t take your opportunities in this competition you are unlikely to get a result. We showed a huge amount of effort from a very young side and we are very proud of that – sometimes you need a little bit more.
“We’re probably out of it now, but we can’t afford to feel sorry for ourselves. This young group of players needs to learn from these experiences. What is also disappointing is that we created a lot of chances, but didn’t finish off enough of them. Some of the rugby we played was as good as we have put together all season.”
The Ospreys are seemingly out of the Pool Five race for the quarter-finals with eight points from four matches. Northampton Saints top the pool on 15 points with Racing just a point behind.
Steve Tandy’s men face Northampton at the Liberty Stadium in January before their final match in Italy against Benetton Treviso. And Biggar was left disappointed after falling just short against the French heavyweights.
He said: “I think we actually edged the game. It was a game of two halves – they were the better side in the first and we had the second. But we’re hugely disappointed because of the tries that they scored and the fact we didn’t take our chances. Racing have a huge pack, we gave them too many turn-overs and the yellow card cost us as well.”
