What is the limit of
$$\lim_{u+v\rightarrow 1}\frac{\ln \int f_0(y)^{1-v} f_1(y)^{v}\mathrm{d}y- \ln \int f_0(y)^u f_1(y)^{1-u}\mathrm{d}y}{1-(u+v)}$$
where $f_0$ and $f_1$ are some density functions.
How to approach this problem? I am confused to find a solution because I cannot simply apply L'Hospital's rule.
I think I must be more clear. Because the limit seems to vary for different $f_i$ and probably it is not possible to give such a limit in terms of $f_i$, if they are unknown.
Is this limit always finite?