  This paper is devoted to the speed (rate) of convergence of Chernoff
approximations to strongly continuous one-parameter semigroups. We provide
simple natural examples for which this convergence: is arbitrary high; is
arbitrary slow; holds in the strong operator topology but does not hold in the
norm operator topology. We also prove general theorem that gives estimate from
above for the speed of decay of the norm of the residual appearing in Chernoff
approximations. We provide also supplementary theorems which makes it easier to
check the conditions of the main theorem. This text is a first draft preprint.
