Taking the limit $$\lim_{p\rightarrow \infty} \left( \frac{\|f\|_\infty}{\|f\|_p}\right)^p$$
First I think the expression after taking the limit will depend on the function $f$.
In my attempt, because it is in the form $``1^\infty"$, I tried to use L'Hopital's rule. And we can calculate the limit (assuming the integrals are defined and finite, I just want to see what the limit might look like). \begin{align*} \lim_{p\rightarrow \infty} \left(\frac{\|\nabla u \|_\infty}{\|\nabla u\|_p}\right)^{p} &= \lim_{p\rightarrow \infty} \exp\left( p\log\left(\frac{\|\nabla u \|_\infty}{\|\nabla u\|_p}\right)\right)\\ &=\lim_{p\rightarrow \infty} \exp\left( \frac{\log\left(\frac{\|\nabla u \|_\infty}{\|\nabla u\|_p}\right)}{\frac{1}{p}}\right)\\ &=\lim_{p\rightarrow \infty} \exp\left( \frac{\frac{d}{dp} \left[-\log\left(\frac{\|\nabla u \|_p}{\|\nabla u\|_\infty}\right)\right]}{\frac{-1}{p^2}}\right)\\ &=\lim_{p\rightarrow \infty} \exp\left( \frac{\left(\frac{\|\nabla u \|_\infty}{\|\nabla u\|_p}\right)\frac{\frac{d}{dp}\left[\|\nabla u\|_p \right]}{\|\nabla u \|_\infty}\}}{\frac{1}{p^2}}\right)\\ \end{align*} where \begin{align*} \frac{d}{dp}\left[\|\nabla u\|_p \right] &= \frac{d}{dp}\left[\left(\int_{\mathbb R^N} |\nabla u |^p dx \right)^{1/p} \right] \\ &=\frac{d}{dp}\left[\exp\left(\frac{1}{p} \log\left(\int_{\mathbb R^N} |\nabla u |^p dx \right)\right)\right] \\ &=\|\nabla u \|_p \left\{\frac{-1}{p^2}\log\left(\int_{\mathbb R^N} |\nabla u |^p dx \right) + \frac{1}{p} \frac{1}{\int_{\mathbb R^N} |\nabla u |^p dx }\int_{\mathbb R^N} |\nabla u|^p \log(|\nabla u |) dx \right\} \end{align*}
Putting it back into the limit we get $$\lim_{p\rightarrow \infty} \exp \left(-\log\left(\int_{\mathbb R^N} |\nabla u |^p dx \right) + p \frac{1}{\int_{\mathbb R^N} |\nabla u |^p dx }\int_{\mathbb R^N} |\nabla u|^p \log(|\nabla u |) dx \right)$$ which simplifies to $$\lim_{p\rightarrow \infty} \frac{\exp \left(p\frac{\int_{\mathbb R^N} |\nabla u|^p \log(|\nabla u |) dx}{\int_{\mathbb R^N} |\nabla u |^p dx } \right)}{\int_{\mathbb R^N} |\nabla u |^p dx }$$
And I am stuck. Is this a correct approach?
Thank you very much!