In a precarious position; also, in a state of keen excitement, as from danger or risk. For example, When the stock market crashed, their whole future was on the edge, or Skydivers obviously must enjoy living on the edge.
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on the edge of. On the point of doing something, as in He was on the edge of winning the election when the sex scandal broke. [ c. 1600 ]
Both def. 1 and 2 allude to the danger of falling over the edge of a precipice.