How to run system diagnostic

It's the D key… (for (D)diagnostic test )
http://support.apple.com/kb/PH14291

 

That would only make sense if you were trying to boot to a system restore DVD (and that DVD was not the disk that came with that Mac)
An Apple restore DVD, and its included diagnostic test, won't run on a Mac different from the system that it shipped with.
And, it's an older Mac, prior to (2010?), which won't boot to the internet test.
http://support.apple.com/kb/TS4154

 

Have you tried holding option + D together immediately after turning on the Mac? It should boot into diagnostic mode

 

Which model MacBook Air do you have?
Year, screen size, and processor would be great - or you can check in the System Profiler, or System Information on a newer Mac.
Get to that quickly from the Apple menu/About this Mac.
Click on the More Info button.
And, post the Model Identifier.
It will be something like MacBookAir3,1
Also, which OS X version are you running now?

According to http://support.apple.com/kb/TS4154
A Mac that did NOT originally ship with Lion may connect to install OS X over the internet, even with an erased drive, no need for any stick. HOWEVER, on a MacBook Air sold before Lion, the Diagnostic test (Restart with D) is only available on the original media (that stick that you have lost). That's also the reason for the message "does not support this machine". So, the only method to run diagnostics on a MacBookAir3,1 would be to find another installer stick. Apple MIGHT be able to provide a replacement, if you contact AppleCare, or ask a "genius"

 

I though that was it.
Check the additional info in my post #7, which should explain what's happening.

 

hmmm I also can't even run system diagnostic on my macbook air 2013 13". Bought with mountain lion preinstalled, now using latest mavericks

 

But - the operating system does not provide the hardware diagnostic test.
The diagnostic test also does not come on older OS X installer DVDs - unless that particular DVD came originally with that older Mac - and the hardware test will ONLY run on that same Mac model
The Diagnostic test is a separate partition on your original installer media - the stick, in your case. That's why you need either that original stick, or one that came with another Late 2010 MBAir - or ask Apple support if a replacement is available.
The original stick is, therefore, the ONLY way to run the Diagnostic test, unless you can find the exact same hardware test. Just for your info, even the 13-inch Late 2010 diagnostic test is a different version and won't run on your 11-inch Late 2010. And, the hardware test version that you want for an 11-inch Late 2010 MBAir is AHT 3A205
And missin that, I would suggest, for a hardware test - take your MBAir to an Apple "genius", for full tests.

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That is incorrect.
Your much newer MBAir will boot to the Apple Diagnostics over the internet, and will assist you if the internet connection is not working properly.
http://support.apple.com/kb/HT5781