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I am trying to solve this question

Let $f:[0,1]\rightarrow \mathbb{R}$ be a continuous function such that $f(0)=2016$. Find $$\lim_{n\to \infty}\int_{0}^{1}f(x^n)dx.$$

I don't know how to approach this problem. Please help.

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    $\begingroup$ I wonder from what year this question is. $\endgroup$
    – user370967
    Jan 1, 2018 at 11:26

1 Answer 1

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$f$ is continuous on $[0,1]$ and thus bounded. Moreover, since $x^n \to 0$ for all $x\in [0,1)$ we have that the integrand converges pointwise almost everywhere to $f(0)$ (by continuity). So we can apply the dominated convergence theorem to get that

$$\lim_{n \to \infty}\int_0^1 f(x^n) \ dx = \int_0^1 f(0) \ dx = 2016.$$

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