  Understanding the asymptotic behavior of physical quantities in the
thermodynamic limit is a fundamental problem in statistical mechanics. In this
paper, we study how fast the eigenstate expectation values of a local operator
converge to a smooth function of energy density as the system size diverges. In
translation-invariant quantum lattice systems in any spatial dimension, we
prove that for all but a measure zero set of local operators, the deviations of
finite-size eigenstate expectation values from the aforementioned smooth
function are lower bounded by $1/O(N)$, where $N$ is the system size. The lower
bound holds regardless of the integrability or chaoticity of the model, and is
saturated in systems satisfying the eigenstate thermalization hypothesis.
