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Jurgen Klopp's side went into the home leg of the tie 3-1 down, having been outplayed in Madrid a week ago, and needed one of their famous nights at Anfield to complete the comeback.
The Liverpool boss admitted before the game that, despite their away goal in the Spanish capital, things would be difficult because the lack of the fans as their 12th man, as usually happens.
The hosts had the best of the play on the night and created several extremely good chances at Thibaut Courtois' goal but their finishing wasn't up to much.
From 15 shots on the Real goal, the Premier League champions could muster just four on target as they drew a blank against the 13 times European champions, who booked their place against Chelsea in the semi-finals.
With the final in Istanbul this year, the scene of Liverpool's famous final victory in 2005, the club will have been desperate to make it back there.
The loss means that Klopp's side will be trophyless for the first time in three seasons, having already been knocked out of both domestic cups and being mathematically out of the title race.
Klopp and his players must now turn their attention on qualifying for next season's Champions League, which is far from guaranteed.
The team are currently three points behind fourth place West Ham in sixth with just seven games of the season to go.
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