How to calculate this integral $I=lim f^n d\mu$

Question

Let X be such that $\mu (X)=1$, and $f$ be a measurable function such that $f:X\rightarrow [0,\infty[$

Calculate $I=lim_{n\rightarrow \infty} \int_X f^n d\mu$ and $I=lim_{n\rightarrow \infty} \int_X atan(f^n) d\mu$.

I don't really know where to start and what to do, I'm a beginner in measure theory. Thank you.

The exercise tells to use the sets where $f$ is less than $1$ and when it isn't.