The most severe impact of climate change on vegetation growth and agriculture is likely to occur under water-limited conditions. Under such conditions the plants optimize the inward flux of CO2 and the outward flux of water vapor (the transpiration) by regulating the size of the stomatal openings. Higher temperature increases water loss through transpiration, forcing the plants to diminish the stomatal openings, which decreases photosynthesis. This is counteracted by higher CO2 concentration, which allows plants to maintain the inward flux of CO2 through the smaller openings. These two counteracting effects, combined with the change in precipitation, determine the net change of biological productivity. Here, a vegetation sensitivity approximation (VSA) is introduced, in order to understand and estimate the combined effect of changed temperature, CO2 and precipitation to first order. The VSA is based on the physical laws of gas flux through the stomatal openings, and is only valid under water-limited conditions. It assumes that the temperature depends logarithmically on the CO2 concentration with a given climate sensitivity. Precipitation is included by assuming that it is proportional to the transpiration. This is reasonable underwater-limited conditions, when transpiration is often a large fraction of the precipitation. The VSA is compared to simulations with the dynamic vegetation model LPJ. The agreement is reasonable, and the deviations can be understood by comparison with Koppen's definition of arid climate: in an arid climate growth increases more according to LPJ than according to the VSA, and in non-arid conditions the reverse is true. Both the VSA and the LPJ simulations generally show increased growth with increasing CO2 levels and the resulting temperature increase, assuming precipitation to be unchanged. Thus, in this case the negative temperature effect is more than compensated by the positive effect of CO2.