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To throw in a curve ball
2x 580 3gb cards are in your budget in the second hand market and they'll do 60fps at 1440p in BF3 easily.
Davepen what CPU do you have and what's it running at?
This could be your problem.
2x680s can cut it, just like Davepen has shown.
Ok the first section, upto and including the ballistia, the archers and the final boss fight. ultra, vsync off:
Frames, Time (ms), Min, Max, Avg
41919, 497596, 28, 121, 84.243
Frames, Time (ms), Min, Max, Avg
13700, 166671, 30, 115, 82.198
It comes down to if you want to maxx the settings or not.
I run 3x7970's (watercooled) with both 1080p/120hz and 1440p/60hz displays.
I like to have a minimum frame rate of 50-60fps at 1440p, 30-60fps don't work for me as I get motion sickness.
If you want the settings maxxed at 1440p, you will need 3 cards, 2 cards won't cut.
You need 2 for 1080p in Battlefield 3 in Ultra.
My own experiences with Skyrim and BF3 is that average frame rates mean nothing, you need to be looking at the minimum frame rate and frame draw times and looking at how long they stay at their minimums, that is why sites like Anandtech can't be used for reference points anymore.