Eliminate derives from Latin ēlīmināre (to push beyond the threshold), from ex- (out) and līmen (threshold), and has broadened from literal banishment to general removal.
To completely remove or get rid of something; to exclude from consideration or competition
Eliminate entered English in the mid-16th century from Latin ēlīminātus, past participle of ēlīmināre, meaning 'to turn out of doors, to banish.' The verb is a compound of ex- (ē-) (out of) and līmen (threshold, doorstep). The original sense was literal: to push someone beyond the threshold, to expel from the house. Latin līmen (genitive līminis)