Limits of Integral $\lim\limits_{n\to\infty}\int\limits_a^b\sqrt[n]{f^n(x)+g^{n}(x)}dx.$ Let $f, g:[a,b]\to[0,\infty)$ be continuous functions. Find the value of $$\lim\limits_{n\to\infty}\int\limits_a^b\sqrt[n]{f^n(x)+g^{n}(x)}dx.$$ 
One said that the results is $\int\limits_a^b h(x)dx$ where $h(x)=\max\{f(x), g(x)\}$. 
Please give me some hints.
 A: Suppose $f(x) > g(x) \ \forall x \in [a, c]$ and $g(x) > f(x)\ \forall x \in [c,b]$
$$
\lim_{n \rightarrow \infty} \int\limits_{a}^b \sqrt[n]{f^n(x)+g^n(x)} \mathrm{d}x \\
\lim_{n \rightarrow \infty} \int\limits_{a}^c \sqrt[n]{f^n(x)+g^n(x)} \mathrm{d}x + \lim_{n \rightarrow \infty} \int\limits_{c}^b \sqrt[n]{f^n(x)+g^n(x)} \mathrm{d}x \\
\lim_{n \rightarrow \infty} \int\limits_{a}^c \left|f(x)\right| \sqrt[n]{1+\frac{g^n(x)}{f^n(x)}} \mathrm{d}x + \lim_{n \rightarrow \infty} \int\limits_{c}^b \left|g(x)\right|\sqrt[n]{\frac{f^n(x)}{g^n(x)}+1} \mathrm{d}x \\
$$
Now take the limit inside the integral sign and you will see that the problem will reduce to
$$
\lim_{n \rightarrow \infty} \int\limits_{a}^c \left|f(x)\right| \mathrm{d}x + \lim_{n \rightarrow \infty} \int\limits_{c}^b \left|g(x)\right| \mathrm{d}x \\
$$
I hope you get the general idea. You split the interval $[a,b]$ into parts wherever $f$ and $g$ cross each other and do a procedure similar to the one above and you will find that the function that is bigger in an interval  will appear in the integrand of the interval.
EDIT: When $f(x) = g(x)$ in an interval (it doesn't matter if that happens only at a countably infinite points), then the integrand will just be $f(x)$ (or $g(x)$) (why?)
Thanks to Glen O for pointing this out.
