Will my MacBook air (mid 2013) deliver 3440x1440 resolution to an external display?

My 2013 macbook air, with thunderbolt port, appears to be incapable of supporting ultra wide screen external displays (3440x1440 resolution).

I am currently using a mini display port to HDMI adaptor, but have *just* ordered a thunderbolt to DisplayPort adaptor, to see whether that works.

Any suggestions - am I trying to solve an problem which can't be solved?