Prove that the following limit exists $\lim_{n\to\infty}\left(\int\limits_{0}^{1}\vert f(x)\vert^{n}dx\right)^{\frac{1}{n}}$
If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?