The Pilot Flying (PF) Elected To Continue To The Planned Destination Airport Analysis: The on-demand air taxi flight proceeded to the destination airport and, about 1 hour and 22 minutes before landing, the flight crew reviewed the destination airport automated terminal information service (ATIS), which indicated the wind was from 270° at 14 kts, gusting to 24 kts. The cockpit voice recorder (CVR) recorded the crew discussing the crosswind component of the winds reported on the ATIS, the aircraft operating limitations, company procedures, and an alternate airport, but the pilot flying (PF) elected to continue to the planned destination airport. About 16 minutes before the airplane touched down, when it was about 39 nautical miles (nm) from the destination airport, the PF advised the pilot monitoring (PM) that he checked the airport’s automated surface observing system (ASOS) and reported the wind velocity was currently at 270° at 13-14 kts. While on final approach, about 1.2 nm from the approach end of runway 36L, the tower controller advised that the wind was from 290° at 19 kts, gusting to 24 kts.