Proving that $\lim_{n\to\infty}(\int_{a}^{b}f(x)^ndx)^{1/n} = \max_{x\in [a,b]}f(x)$
Finding $\lim\limits_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$ for continuous $f:[0,1]\to[0,\infty)$