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I am reading the following paper 1998(H.Hudzik) P.574

It reads using L'Hopital rule$$\liminf_{u\to\infty} \frac{1/\varphi(1/u)}{\psi(u)}=\liminf_{u\to\infty}\frac{\varphi'(u)}{\psi'(u)u^2[\varphi(1/u)]^2}.$$ That means we can apply L'Hopital for lower limits i.e. $$\liminf_{u\to\infty} \frac{f(u)}{g(u)}=\liminf_{u\to\infty}\frac{f'(u)}{g'(u)}?$$ But I only know the classical one. Is there someone can give me some reference to check this formula? Or if possible someone can give a proof?

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The full L'Hopital rule says that $$\liminf \frac{f'}{g'}\leq\liminf\frac{f}{g}\leq\limsup\frac{f}{g}\leq\limsup\frac{f'}{g'}.$$ So in the special case when the limit of $f'/g'$, exists, the limit of $f/g$ also exists and is equal to the limit of $f'/g'$.

This general rule is proved by integration.

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    $\begingroup$ For those interested, a little known paper of William Henry Young, On indeterminate forms [Proceedings of the London Mathematical Society (2) 8 (1910), pp. 40-76], extended this by proving (among other things) that every subsequential limit point of $f/g$ is a subsequential limit point of $f'/g'.$ In fact, I think Young's paper is the first paper in which even the weaker version involving $\liminf$ and $\limsup$ appears, the weaker version being (I think) rediscovered by Picone (1929) (continued) $\endgroup$ Jan 18 at 17:40
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    $\begingroup$ and Amante (1936) and Halvorsen (1963). A nice survey of many subtle versions of L'Hopital's rule is given in: Ioan Muntean, L'Hôpital's rules with extreme limits , pp. 11−28 in Seminar on Mathematical Analysis (Preprint 93-7), "Babeş-Bolyai" University, Cluj-Napoca, 1993 (MR1333230; Zbl 833.26007). Although I have a photocopy of this paper, I cannot find it on the internet. Muntean cites Picone's paper, but not the papers by Young or Amante or Halvorsen. $\endgroup$ Jan 18 at 17:57
  • $\begingroup$ Thank you very much for your nice reference. This is much helpful. $\endgroup$ Jan 19 at 8:05
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Q: Is it true that $$\liminf_{u\to\infty} \frac{f(u)}{g(u)}\stackrel{?}{=}\liminf_{u\to\infty}\frac{f'(u)}{g'(u)}.$$ A: No, try $f(u)=u+\sin u$ and $g(u)=u$, then the left-hand-side of the equation equals 1, while the right-hand-side equals 0. The best you can do is replace = by $\geq$.

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  • $\begingroup$ But my confused point why this paper use this rule, if we want this formula equal, which condition should we consider? $\endgroup$ Jan 18 at 10:55
  • $\begingroup$ the equality is correct if $\lim_{u\rightarrow\infty}f'(u)/g'(u)$ exists. $\endgroup$ Jan 18 at 11:29

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