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I got stuck on three limit problems. Could anyone give me a hint about them?

Let $f\in C([0,1])$, find the following limits:
(1)$\lim_{n\to \infty}\int_0^1 x^nf(x)dx$;
(2)$\lim_{n\to \infty}n\int_0^1 x^nf(x)dx$;
(3)$\lim_{h\to 0^+}\int_0^1 \frac{h}{h^2+x^2}f(x)dx$.


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Effort, self work, ideas...? – DonAntonio Mar 10 '13 at 22:54
Do you know the dominated convergence theorem? – Ayman Hourieh Mar 10 '13 at 22:56
up vote 0 down vote accepted

I want to give you the hints and my results for the three, but try to finish the proof by your self.

  1. $|f|\le M$, and $\int_0^1 x^n \mathrm{d}x=\frac{1}{n+1}$
  2. $$n\int_0^1 x^nf(x)\,\mathrm{d}x=n\int_{(0,\,1-\delta)} x^nf(x)\,\mathrm{d}x+n\int_{(1-\delta,\,1)} x^nf(x)\,\mathrm{d}x$$ then prove $$n\int_{(0,\,1-\delta)} x^nf(x)\,\mathrm{d}x \to 0$$ and $$n\int_{(1-\delta,\,1)} x^nf(x)\,\mathrm{d}x\to f(1)$$
  3. $$\int_0^1 \frac{h}{h^2+x^2}\cdot f(x)\,\mathrm{d}x=\int_{(0,\,\delta)} \frac{h}{h^2+x^2}\cdot f(x)\,\mathrm{d}x+\int_{(\delta,\,1)} \frac{h}{h^2+x^2}\cdot f(x)\,\mathrm{d}x$$ then prove $$\int_{(0,\,\delta)} \frac{h}{h^2+x^2}\cdot f(x)\,\mathrm{d}x\to \frac{\pi}{2}\cdot f(0)$$ and $$\int_{(\delta,\,1)} \frac{h}{h^2+x^2}\cdot f(x)\,\mathrm{d}x\to 0$$
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