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 $f$ such that $f\in L^p(\mu)$ for all $1\leq p<\infty.$ (Although I only really need it for $k=2.$) Note that these limits are related by L'Hospital's rule.