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If $\displaystyle \lim_{x \to \infty} f(x) = a$, and knowing that $\displaystyle \lim_{x \to \infty} xf'(x)$ exists , how would I find that limit?

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up vote 8 down vote accepted

I think I've seen this question before, but I've thought of a pleasant little proof.

$\lim f(x) < \infty$, so $\lim \dfrac{f(x)}{\ln x} = 0$

But $\lim \dfrac{f(x)}{\ln x} = \lim \dfrac{f'(x)}{\frac{1}{x}} = \lim x f'(x)$ if $\lim x f'(x)$ exists.

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