According the mathematics of pharmacological half-lives, shouldn't a drug NEVER be completely purged from?

No, that's not true. A pharmacological half-life is the time it takes for a substance to lose half of its pharmacologic activity. There will come a point in the decay process that there in an insufficient amount of the substance left to sustain activity. Decay processes that are often treated as exponential, are really only exponential so long as the sample is large and the law of large numbers holds.