I would like to find the convergence rate of the following function:
$$f(x) = |e^{-ax}-e^{-bx}|,$$
with $a,b>0$ and $x\to+\infty$. By finding the convergence rate, I mean finding the largest possible $c>0$ such that
$$f(x)\le k e^{-cx}\quad \forall x\ge x_0$$
for some $k>0$ and $x_0\in\mathbb{R}$. Of course the inequality holds for $c:=\min\{a,b\}$; but is that the largest possible $c$? And if so, how can we show that?
Assume without loss of generality that $0<a<b$. Then $$ |e^{-ax}-e^{-bx}|=e^{-ax}\,\bigl|1-e^{-(b-a)x}\bigr|. $$ Since $b-a>0$, we have $$ 1-e^{-(b-a)}\le1-e^{-(b-a)x}\le1,\quad x\ge1. $$