In this paper we consider a general family of EWMA charts for an arbitrary parameter of the target process. Our assumptions on the target process are very weak and they are usually satisfied if it is stationary. We distinguish between the EWMA chart based on the exact variance and the EWMA scheme based on the asymptotic variance. In the case of the EWMA chart with exact variance the in-control variance of the EWMA recursion at time t is used for the decision at time t while in the case of the asymptotic variance at each time point the limit of the in-control variance of the EWMA chart for t tending to infinity is applied. It is analyzed how the distributions of the corresponding run lengths behave if the smoothing parameter tends to zero. We show that the distribution of the run length of the EWMA chart based on the exact variance converges to the distribution of the run length of the repeated significance test while the limit of the EWMA scheme based on the asymptotic variance is degenerate. It is either 0 or 1. This result underlines the weakness of the schemes based on the asymptotic variance if the smoothing parameter is small. Moreover, several properties of the limit chart, i.e. the chart based on the repeated significance test, are presented as well.