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> Let f(x) be continuous on [0,1]. Calculate $\lim\_{n \to \infty} n\int\_0^1 x^n f(x)dx$. 
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What immediately jumps out at me is how close $\frac{x^nf(x)}{\frac{1}{n}}$ looks to a derivative, i.e. if I only had the function $$\frac{(x+\frac{1}{n})^nf(x+\frac{1}{n})-x^nf(x)}{\frac{1}{n}}$$ inside the integrand, then I could answer the question with $1^nf(1)-0^nf(0)=f(1)$. So I am at a dead end down this path. 

I've also tried using the Lebesgue DCT by wondering whether the series of functions $nx^nf(x)\to 0$ pointwise on $[0,1)$ since $f(x)$ is continuous and thus bounded on $[0,1]$. 

Another attempt involved trying to use the second MVT, but that doesn't succeed, either, since I do not know beforehand that the value $k \in [0,1]$ such that $$n\int\_0^1 x^n f(x)dx=f(k)\int\_0^k nx^ndx$$ is actually $k=1$. Any hint would be appreciated!