Communication Breakdown; Distraction; Situational Awareness
Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence
Air Traffic Control Provided Assistance; Flight Crew Became Reoriented
Human Factors; Weather
ATIS had winds 040/7 360v150. We were approaching 28 for departure when Tower called traffic on four-mile final; gave us the winds and cleared us for takeoff. I heard the winds; but they didn't register in my consciousness. We took off; and used a lot of the runway. On climb out above 10;000'; we asked each other about the winds; and pieced together that they were stronger out of the east than we had realized. We got the new ATIS; which had just changed; and copied winds of 070/10G16. Entering that into the [flight computer]; we saw that was 13 knots of tailwind; and out of limits.
[Report narrative contained no additional information.]
B737 flight crew reported taking off with a tailwind out of limits after having been given the latest winds by the Tower.