In the $R_\xi$ gauge, the Feynman rule for the propagator of a massive vector boson is

\begin{equation} D_{\mu\nu} = \frac{i}{k^{2}-M^{2}+i\varepsilon}\left[-g_{\mu \nu}+(1-\xi) \frac{ k_{\mu} k_{\nu}}{k^{2}-\xi M^{2}}\right]. \end{equation}

If we go to the unitary gauge using $\xi = \infty$, the propagator would read

\begin{equation} D_{\mu\nu}^{\xi = \infty} = i\frac{-g_{\mu \nu}+ \frac{ k_{\mu} k_{\nu}}{M^{2}}}{k^{2}-M^{2}+i\varepsilon}. \end{equation}

However, we may have this object inside a loop, that is, there may be an integration over $k$ for all momenta if we want to compute a one-loop diagram.

In case I would want to compute that diagram in the unitary gauge, is it legitimate to take the limit $\xi \rightarrow \infty$ to put the propagator in the form above?

If not, what is the right thing to do?