Intersections of functions with monotone derivatives

Question

Suppose $I=(a,\infty)\subset\mathbb{R}$ and $f,g\in C^{\infty}(I,\mathbb{R})$ with the property that $f^{(n)}$ and $g^{(n)}$ are monotone for all $n\in\mathbb{N}.$ I'm interested in the set of points where $f$ and $g$ coincide: $S=\{x\in I : f(x)=g(x)\}.$ Specifically, is it possible for $S$ to be unbounded but $\limsup\limits_{x\to\infty} |f(x)-g(x)| = \infty$?

Answer 1

Try $a=0$, $f(x) = \exp(2 x)$, $g(x) = \exp(2 x) + c \exp(x) \sin(x)$ where $0 < c < 1$.