A difficulty in understanding a proof for L'Hospital's rule (in Petrovic)

Question

The theorem and its proof is given below:

But I could not understand why $F$ & $G$ are defined as thought, could anyone explain this for me please?

Answer 1

The author has already proved L'Hopital's rule when the limit is taken as a point $a\in\mathbb R$. In order to be able to use that, he defined $F$ and $G$ as he did because$$\lim_{x\to0}\frac{F(x)}{G(x)}=\lim_{x\to\infty}\frac{f(x)}{g(x)}.$$

Answer 2

They are defined that way so that the limit $$\lim_{x\to\infty}F(x)=\lim_{t\to 0^+} f(t)$$

and you can then use the L'Hospital theorem for limits where $x\to a$ and $a\neq \infty$ (in particular, you use the theorem where $x\to 0$).