Show that $\lim_{n\to \infty}\left[ \int_a^bf(x)^n\,dx \right]^{1/n}=\max_{x\in[a,b]}f(x).$
If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?