I have the old MacBook Air with the software OS X 10.8.3 (12D78) and I am unable to connect to my external Apple Monitor. My laptop does not even pick it up. Please help

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If you have a Late 2010 or older MacBook Air and you have a Thunderbolt Display, note that this display is not compatible with your MacBook Air. The first MacBook Air with Thunderbolt was the Mid 2011 one.

If you want an Apple monitor compatible with your Mac, you need a LED Cinema Display, or get a third-party monitor with its adapter to connect it to your MacBook