I am reading the following paper [1998(H.Hudzik)](https://www.jstor.org/stable/44152979) 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?



L'Hopital rule for upper and lower limit?
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.


