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Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a differentiable function whose derivative is continuous. Then, find $$ \lim_{n \to \infty} (n+1) \int_{0}^1 \! x^n f(x) dx $$ I know how to compute limits of integration using l hospital rule. In this problem, I am not to proceed with the problem. Any help will be appreciated. Thanks in advance.

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HINT: Use integration by parts and the fact that $|f'(x)|\le M$ (for some constant $M$) on $[0,1]$.

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  • $\begingroup$ I got this after using by part. \lim_{n\rightarrow \infty}[\int_{0}^{1} f(x) x^{n+1}dx]-\lim_{n\rightarrow \infty}[\int_{0}^{1} f'(x) x^{n+1}dx]. But I am not able to get the fact that you are talking about. Is M is Maximum between the limits of integration. $\endgroup$ – RAHUl JHa Jan 3 '18 at 7:58
  • $\begingroup$ The first term does not have an integral in it! Yes, you're right about $M$; just give an upper bound for the second term and show it goes to $0$. $\endgroup$ – Ted Shifrin Jan 3 '18 at 8:01
  • $\begingroup$ Ok, thanks I got your explanation. You explained nicely. $\endgroup$ – RAHUl JHa Jan 3 '18 at 8:06
  • $\begingroup$ Glad to help. Enjoy learning mathematics! $\endgroup$ – Ted Shifrin Jan 3 '18 at 8:12

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