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Let $g:[0,1]\mapsto\mathbb{R}$ be a continuous function, and $\lim_{x\to0^+}g(x)/x$ exists and is finite. Prove that $\forall f:[0,1]\mapsto\mathbb{R}$,

$$\lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx=f(1)\int_0^1\frac{g(x)}{x}dx$$

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    Where exactly are you stuck? (Also, I think $f$ should be continuous as well).2012-11-16
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    Or we can use the substitution $t=x^n$ to get a clearer view of what is going on as $n\to\infty$.2012-11-16

2 Answers 2