How to take the limits of the sandwiched Renyi divergences?

Question

The sandwiched Renyi divergence is defined as

$$\begin{equation} \tilde{D}_{\alpha}(\rho \| \sigma):=\frac{1}{\alpha-1} \log \operatorname{tr}\left[\left(\sigma^{\frac{1-\alpha}{2 \alpha}} \rho \sigma^{\frac{1-\alpha}{2 \alpha}}\right)^{\alpha}\right]. \end{equation} $$

One can define the limiting cases as $\alpha\rightarrow 1$ and $\alpha\rightarrow\infty$ to obtain the relative entropy and max-relative entropy respectively. I am not sure how to take these limits - does one apply L'Hôpital's rule and if yes, how should one deal with the terms inside the trace? The goal is to show

$$\lim_{\alpha\rightarrow 1}\tilde{D}_{\alpha}(\rho \| \sigma) = \text{tr}(\rho\log\rho - \rho\log\sigma)$$ and

$$\lim_{\alpha\rightarrow \infty}\tilde{D}_{\alpha}(\rho \| \sigma) = \inf \{\lambda: 2^\lambda\sigma\geq \rho \}$$

Answer 1

Both limits are dealt with in a fair amount of detail in the work that originally defined the sandwiched entropies: On quantum Renyi entropies: a new generalization and some properties. In particular, you'll find the relevant results in section IV.C.