Does lim (t to infty) E(x-x hat) = 0 imply that there will be less disturbance over time?

$E(x)\,$ here denotes the expected value of $x\,$, where $x\,$ is a random variable. $\lim _{t\to \infty }E(x-{\hat {x}})=0\,$ only implies that the mean estimation error will converge to zero over time. Disturbance, however, will affect the variance of the error, which is given by $E(x-{\hat {x}})^{2}\,$. In CDS 110b, we will learn how to design observers that minimize this estimation variance if the disturbance can be modeled as Gaussian noise.

--Shuo