# Limes superior and inferior inequalities with functions and its derivatives

#### drani

Hi, I have got some troubles with Lemma 2. from N.H. Duu, On The Existence of Bounded Solutions for Lotka- Volterra Equations, Acta Mathematica Vietnamica 25(2) (2000), 145-159.

Lemma. Let $$\displaystyle G(t)$$ and $$\displaystyle F(t)$$ be two differentiable functions defined on $$\displaystyle (0,âˆž)$$ such that $$\displaystyle \lim_{tâ†’âˆž}G(t)=\lim_{tâ†’âˆž}F(t)=+âˆž$$ then
$$\displaystyle \limsup_{tâ†’âˆž}\frac{G(t)}{F(t)}â‰¤\limsup_{tâ†’âˆž}\frac{Gâ€²(t)}{Fâ€²(t)}; \liminf_{tâ†’âˆž}\frac{G(t)}{F(t)}â‰¥\liminf_{tâ†’âˆž}\frac{Gâ€²(t)}{Fâ€²(t)}.$$

Proof. By the Cauchy theorem for differentiable functions, for any $$\displaystyle t_1,t_2>0$$ there is a $$\displaystyle \thetaâˆˆ(t_1,t_2)$$ such that
$$\displaystyle \frac{Gâ€²(\theta)}{Fâ€²(\theta)} =\frac{G(t_1)âˆ’G(t_2)}{F(t_1)âˆ’F(t_2)} =\frac{G(t_2)}{F(t_2)}\times\frac{1âˆ’\frac{G(t_1)}{G(t_2)}}{1âˆ’\frac{F(t_1)}{F(t_2)}}.$$
Letting $$\displaystyle t_1$$ and $$\displaystyle t_2\to\infty$$ such that $$\displaystyle \lim\frac{G(t_1)}{G(t_2)}=\lim\frac{F(t_1)}{F(t_2)}=0$$ we get the result.
Can anybody help me understand the last part of this proof? Exactly, how this transformation using Cauchy theorem with included convergences leads to inequalities from lemma?

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#### mathman

Forum Staff
It looks like $t_2 \to \infty$ first.