Can 2013 MacBook Air output to 2560x1080? I recently purchased a 21:9 LG monitor (model 29UM57) and I'm having difficulty getting my 2013 MBA to output to 2560x1080, which is the monitor's resolution. My max resolution is 1080p.

I purchased this MDP-HDMI adapter, Cable Matters® Gold Plated Mini DisplayPort (Thunderbolt™ Port Compatible) to HDMI Male to Female Adapter - 4K Resolution Ready, hoping that it would help my situation, but my max resolution has not changed.

I'm wondering whether ordering this HDMI cable, AmazonBasics High-Speed HDMI Cable - 6.5 Feet (2 Meters) Supports Ethernet, 3D, 4K and Audio Return, will fix my problem. I don't know what version or whatever my current HDMI cable is.

Please advise.
I believe the problem is your HDMI cable - it's not capable of displaying a resolution of 2560x1080.  In other words, it's pre-HDMI 1.4 specification.  (Also, the link provided to the cable in the original question is out of date; it's no longer available).

Per Apple Support, the 2013 MacBook Air (11" and 13" model) has a Thunderbolt port which also supports mini Display Port.



Since the HDMI cable referenced in the original question is 4K capable (HDMI v1.4 specification), the cable is not the issue.  What wasn't mentioned was the HDMI adapter. I believe this component to be the "weak link" in this chain because if the mDP to HDMI adapter isn't 4K capable, nothing else will matter.

I recommend getting an active mDP to HDMI adapter (rather than passive) and then connecting your HDMI (v1.4 or above) cable from the adapter to the TV.   



That said, I would only recommend this solution for this particular setup, because the LG monitor 29UM57-P only supports HDMI connections.

Ideally, you would want to use mDP to either mDP or DP connections rather than convert to HDMI, DVI, etc., to get the most reliable connection(s).  Less conversions means less points of potential failure.