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I have been asked to prove that if $f,g:[a,b]\to[0,\infty)$ are continuous functions, then $$\lim_{n\to\infty}\int_a^b\sqrt[n]{f^n(x)+g^n(x)}dx=\int_a^b \max\{f(x),g(x)\}dx$$

but I have no idea how to due it, could someone show a step by step process all the way to the answer?

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    The Squeeze theorem should help. – 2017-01-03
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    Yes, I tried this, the upper limite is simple, but I didn't find a way to find the lower limite. – 2017-01-03

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As the comment pointed out that you should use $\text{max}(f(x),g(x))^n < f(x)^n+g(x)^n< 2\text{max}(f(x),g(x))^n$, and taking root and integrate both sides to yield the answer.