Find $\lim\limits_{n\rightarrow\infty}\int\limits_0^1f(x^n)dx$

Okay, so please let me state an analysis principle that will be very useful in numerous problems.

Interversion is painful.

Either there is a straightforward (« logically tautological », that is, that does not require any analysis) argument for doing it, or it requires thinking the whole reasoning through with more elaborate arguments (such as some form of uniform convergence).

So now, let $M$ be a bound for $f$. Let $\epsilon >0$. I suggest that you prove that

$$\left|\int_0^1{f(x^n)}-f(0)\right| \leq \left|\int_0^{1-\epsilon}{f(x^n)}-(1-\epsilon)f(0)\right| + M\epsilon + \epsilon |f(0)|. $$

Thus for every $n$ large enough $$\left|\int_0^1{f(x^n)}-f(0)\right| \leq 3M\epsilon$$.