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Swansea started brightly and had two good chances in quick succession. Angel Rangel's pass found Nathan Dyer, whose cutback fell to Pablo Hernandez. However, the Spaniard shot well over the bar from a good position.
Just a minute later, Michu - who has scored 17 league goals this season - volleyed wide from six yards out.
Swansea had a penalty shout turned down as the half wore on, but the referee decided that Matija Nastasic's challenge on Michu wasn't a foul.
Man City had much more of the ball at the start of the second half, and had claims for a penalty themselves when Dzeko was brought down by Chico Flores, but the referee once again waved play on.
Dzeko, who replaced Yaya Toure at the break, then wasted a glorious chance for Man City. Zabaleta cut the ball back into the path of the Bosnian striker, but he couldn't adjust his feet in time and the ball squirmed wide of the post with the goal gaping.
Just a minute later, Dzeko's blocked effort fell at the feet of Silva at 10 yards, but the Spaniard bent his effort wide of the post.