Originally, to exterminate something was to banish it or drive it away. And it is this meaning that can be found in the Latin origin of "exterminate." "Exterminate" comes from "exterminatus," the past participle of exterminare, meaning "to drive beyond the boundaries." The Latin word exterminare was formed from the prefix ex- ("out of" or "outside") and "terminus" ("boundary"). Not much more than a century after its introduction to English, "exterminate" came to denote destroying or utterly putting an end to something. And that's the use with which the word is usually employed today.