Bi-phasic or diauxic growth is often observed when microbes are grown in a chemically defined medium containing two sugars (for example glucose and lactose). Typically, the two growth stages are separated by an often lengthy phase of arrested growth, the so-called lag-phase. Diauxic growth is usually interpreted as an adaptation to maximise population growth in multi-nutrient environments. However, the lag-phase implies a substantial loss of growth during the switch-over. It therefore remains unexplained why the lag-phase is adaptive. Here we show by means of a stochastic simulation model based on the bacterial PTS system that it is not possible to shorten the lag-phase without incurring a permanent growth-penalty. Mechanistically, this is due to the inherent and well established limitations of biological sensors to operate efficiently at a given resource cost. Hence, there is a trade-off between lost growth during the diauxic switch and the long-term growth potential of the cell. Using simulated evolution we predict that the lag-phase will evolve depending on the distribution of conditions experienced during adaptation. In environments where switching is less frequently required, the lag-phase will evolve to be longer whereas, in frequently changing environments, the lag-phase will evolve to be shorter.