How to evaluate $\lim_{n\rightarrow\infty}\int_{0}^{1}x^n f(x)dx$, 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.

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