# Finding limits with integral sign

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.

HINT: Use integration by parts and the fact that $|f'(x)|\le M$ (for some constant $M$) on $[0,1]$.
• 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 $0$. – Ted Shifrin Jan 3 '18 at 8:01