Does Brans-Dicke theory always yield general relativity in the infinite ω limit?

In this work it is shown that the Brans-Dicke theory of gravity does not always go over to general relativity in the $\omega \to \alpha$ limit. It is also shown, by an order of magnitude estimate, that one can get general relativity as a limit of Brans-Dicke theory when $\omega \to \alpha ,$ only if the trace of the energy-momentum tensor describing all fields other than the Brans-Dicke scalar field is nonzero.