Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit
$$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$
exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)

Moreover generally I would like to know if in general,
$$\lim_{p\rightarrow\infty}\frac {\frac{d^k}{dp^k} \ln \|f\|_{L^p(\mu)}}{\frac{d^k}{dp^k} \ln p}$$
exists in $[0,\infty]$ for any $f,$
although I only really need it for $k=2.$