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?