Can't solve the following problem:

I have two real continuous and differentiable functions: f(x) and g(x); both functions are always strictly decreasing.

I know that lim f(x)=lim g(x)= lim f(x)/g(x)=0, for x->infinity

F(x) and G(x) are the inverse functions of f(x) and g(x); i.e, F(f(x))=x and G(g(x))=x.

Can be proved that lim F(y)/G(y)=0, when y->0?

I have a partial proof for some functions, but i cannot prove it in general.

Thank you.