In a manner that carefully looks at or studies something closely; with careful scrutiny.
From examining (present participle of examine) + -ly (Old English -lice, 'in the manner of'). The adverb form transforms the action into a description of how something is done.
Adverbs ending in -ingly are quirky: they add an extra 'ing' before the -ly, so you get 'examinine' + 'ly' rather than just 'examinedly'—this doubled participle form is a remnant of how Middle English combined different grammatical elements.
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