I have a function and I'm looking for

I know the following about the functions g and h:

We have . At , both numerator and denominator are 0. Applying l'Hopital's rule, we obtain

- and so
- for , for and for
- and finite at
- finite at

.

Applying l'Hopital's rule again, we're back to where we started:

There's probably a simpler way of presenting this problem, sorry about that... If anyone knows how this kind of circularity problem is called and/or how to deal with it, I will be very grateful!