And lo, the spirit and awkwardness of Martin O’Neill’s Republic of Ireland were not enough. Those qualities took them to within touching distance of their first World Cup in 16 years, but no further. Denmark, far superior in this play-off, will go instead. In Christian Eriksen they have a player who belongs among the elite, a point he expressed beautifully in Dublin with an exquisite hat-trick.

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Fleetingly Ireland had Russia within their reach, as Shane Duffy headed them into an early lead. But by half-time the Danes had wrenched the World Cup ticket off the home side and they never gave it back. Age Hareide’s side scored the away goal the Irish had dreaded when Andreas Christensen’s ugly shot found the net via a post and Cyrus Christie’s leg in the 29th minute. Then came Eriksen’s trio of beauties. It seemed like an unnecessary insult when Nicklas Bendtner added another from a late penalty.

And yet this could have been worse for Ireland. When challenged to be more enterprising they became more ragged. Perhaps lack of attacking practice had made them rusty. Several of Denmark’s goals had their origins in Irish blunders. But Darren Randolph made several good saves to prevent a heavier toll. The Danes were a match for Ireland physically and a class above them in terms of skill, fluency and ambition.

While Eriksen sparkled, the closest player that Ireland have to someone of that calibre, Wes Hoolahan, began on the bench again. O’Neill made only one alteration to the line-up that had shut out the Danes in Copenhagen on Saturday, with David Meyler replacing Callum O’Dowda. If Ireland were to attack more adroitly than they did on Saturday, then that change would have to come from the same personnel, with loud verbal cues from a rambunctious crowd.

It took six minutes for that crowd’s wish to be granted. The goal owed nothing to new-found intricacy but no one cared about aesthetics when Nicolai Jorgensen’s attempt to clear a long free-kick by Robbie Brady resulted in the striker looping the ball backwards in his own box, inviting Duffy to soar high and nod the ball past the advancing Kasper Schmeichel and into the net from eight yards.

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Duffy had scored a similar goal in Georgia in the group stages, after which Ireland had become so passive they almost enticed the opponents to equalise, which Georgia eventually did. There would be serious jeopardy in doing the same here. But soon, whether by Ireland’s choice or the duress applied by Denmark, the home side began backpedaling.

They were fortunate in the 10th minute that Jens Stryger Larsen failed to connect properly with the ball after Simon Kjaer caught the Irish defence out with a long diagonal pass. Then they were grateful to Randolph for diving to his left and tipping away a drive from 20 yards by William Kvist. Two minutes later the goalkeeper came to the rescue again, parrying a blast by Pione Sisto.

In an increasingly frenetic game it was not one-way traffic. Daryl Murphy almost doubled Ireland’s lead with a dainty flick at the near post from a cross by Christie, his shot missing the target by inches. Moments later James McClean stormed down the left and into the box before firing a yard past the far post.

Despite those raids, it was not looking like Ireland would last 90 minutes without conceding. As it turned out, they did not even get past 29. There was ingenuity and cruelty in the way they were infiltrated. After a short corner Sisto nutmegged Harry Arter and skittered into the box before sending a low cross to Christensen. The defender scuffed his shot but it hit a post and rebounded in off Christie. As an away goal, the equaliser put Denmark in front.

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Three minutes later the visiting side took a conventional lead, Eriksen sweeping the ball regally into the net from 20 yards via the underside of the crossbar following a move that began when Stephen Ward was dispossessed on the left.

By half-time Ireland knew that to resurrect their World Cup dream they would have to score two goals in 45 minutes and avoid conceding again. Mission impossible.

Now O’Neill adopted a resolutely attacking stance, introducing Hoolahan and Aiden McGeady for Meyler and Arter at half-time. That left a void in midfield and Denmark revelled in it. After a couple more saves by Randolph, the goalkeeper was rendered helpless in the 63rd minute by the sheer brilliance of Eriksen, who crowned a fine collective move by curling a shot into the bottom corner from 20 yards.

Facebook Twitter Pinterest Seamus Coleman consoles an emotional Robbie Brady (10) after the final whistle. Photograph: Crombie/INPHO/Rex/Shutterstock

The Tottenham Hotspur player helped himself to a third goal by punishing another mistake by Ward and ramming the ball into the roof of the net 10 minutes later. Bendtner then scored from the spot after a clumsy tangle with Duffy.