I met "ambit" for the first time in Robert Harris' Enigma, as a word in a Times crossword puzzle:

"'Morning snack as far as it goes.' Five letters."
"Ambit."
"How do you get it so quickly?"
"It's not hard. You learn to know the way they think. Morning--that's A.M., obviously. Snack as far as it goes--bite with the e missing. As far as it goes--well, within ones ambit. One's limits.

pages 91-92.

I've always pictured it as meaning the distance one can walk, i.e. ambulate