How to find $s(\exp(d(x)))$ ~ $ x + 2 $?

Question

Let $x$ be a positive real. I want to find a pair of analytic functions $s(x),d(x)$ such that $s(d(x)) = x$ and

$ s(\exp(d(x)))$ ~ $ x + 2 $

More presicely I Also want :

$$ \lim_{x \to \infty} s(\exp(d(x))) - x - 2 = 0$$

Polynomials seem to fail as do polynomials of exp or ln.

Maybe try Lambert-W ?

Or do I want THE impossible ?

Answer 1

We have: $$ d(x+2)= e^{d(x)} \tag{1}$$ hence assuming $d(0)=1$ we have that $d$ grows pretty fast: $d(2)=e,d(4)=e^e,d(6)=e^{e^e}$.

Now it is time to look at Anix' answer to this MO question.