If $\det(\Lambda_0) \to 0$, what does $$ \exp\left(-\frac{1}{2}\text{trace}\left(\Lambda_0 \Sigma^{-1}\right)\right)\det\left(\Lambda_0\right)^{-1/2} $$ approach?
I was trying to answer the following question, and I couldn't quite finish it off. It asks for a derivation of why the Jeffreys prior for a multivariate normal distribution is proportional to $\det(\Sigma)^{-(d+1)/2}$, where $d$ is the size/length of an observation.