# L'Hopital rule for upper and lower limit?

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?

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 $$f'/g'$$.
• 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) Jan 18 at 17:40
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 $$\geq$$.
• the equality is correct if $\lim_{u\rightarrow\infty}f'(u)/g'(u)$ exists. Jan 18 at 11:29