極める

also 決める

1. ichidan verb, transitive to decide; to choose; to determine; to make up one's mind; to resolve; to set one's heart on; to settle; to arrange; to set; to appoint; to fix
2. to clinch (a victory); to decide (the outcome of a match)
3. to persist in doing; to go through with
4. as 決めている to always do; to have made a habit of
5. to take for granted; to assume
6. to dress up; to dress to kill; to dress to the nines
7. to carry out successfully (a move in sports, a pose in dance, etc.); to succeed in doing
8. martial arts, sumo to immobilize with a double-arm lock (in sumo, judo, etc.)
9. to eat or drink something; to take illegal drugs

きわめる

1. ichidan verb, transitive to carry to extremes; to go to the end of something; to reach the limits of something
2. to reach the peak of something (e.g. luxury, hardships, etc); to be extremely (e.g. busy)
3. to exhaust completely; to have nothing left (e.g. to say)