Forcing Intel HD Graphics 5000 on a 2013 MacBook Air to output 4K
No, your 2013 MacBook Air cannot drive a 4K display. MacBook Air 2013 comes with a Thunderbolt 1 controller which supports DisplayPort 1.1. DisplayPort 1.2 is needed to get 4K support.

MacBook Air (13-inch, Mid 2013) - Technical Specifications


Graphics and Video Support

Intel HD Graphics 5000.
Dual display and video mirroring: Simultaneously supports full native resolution on the built-in display and up to 2560 by 1600 pixels on an external display, both at millions of colors.



Use 4K displays, 5K displays, and Ultra HD TVs with your Mac


You can use 4K displays and Ultra HD TVs with these Mac computers:

MacBook Air (Early 2015) and later.