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How to evaluate $\lim\limits_{n\rightarrow\infty}\int_{0}^{1}x^n f(x)dx$, well, i did one problem from rudins book that if $\int_{0}^{1}x^n f(x)dx=0\forall n\in\mathbb{N}$ then $f\equiv 0$ by stone weirstrass theorem. please help me here.

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up vote 3 down vote accepted

Assuming $f$ is integrable on $[0,1]$, note that $$\int |x^kf(x)|dx\leq\int |f(x)|.$$ This implies $f(x)$ is finite a.e., hence $x^nf(x)\to0$ as $n\to\infty$. Applying Lebesgue's Dominated Convergence Theorem implies $$\lim_{n\to\infty}\int_0^1x^nf(x)dx=0.$$

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+1 Nice approach. ;-) – Babak S. Jan 16 '13 at 14:25

Assuming $f$ is integrable you can use dominated convergence. If $f$ is positive, monotone convergence works too. In either case, the limit is $0$.

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