what does "limelit air"mean?
To his audiences his easy air seemed unstudied. Very few knew how hard he had toiled to achieve it. On a rostrum the illusion of spontaneity was almost perfect; only his hands would have betrayed him, and he was careful to keep them out of sight. They weren't out of Valenti's sight, though. They were just above him, vibrating so violently at times that they seemed palsied. Now and then the right hand would shoot up and out, the index finger stabbing the limelit air to make a point.

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Answer

The word "limelit"  is an adjective form of the noun "limelight".
Many years ago, in the early days of stage theater, a limelight was a type of lamp used in stage lighting, in which light is produced by heating lime to white heat.
That kind of lighting is no longer used, but the idiom "in the limelight" derives from it, which is to be in a position of public attention or notice (as were the performers on those stages in days gone by).
So the speaker in this paragraph is someone who is well known; someone who is "in the limelight" in his public life. Thus the air around him when he speaks is "limelit air".

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