Inequality including Big-Oh-notation

Question

For positive a,b, I have that $x‘(t)=e^{-ax(t)}-e^{-ay(t)}+O(e^{-(a+b)x(t)})$ and can show that this has the lower bound $x‘(t)\geq O(e^{-(a+b)x(t)})\geq O(e^{-aM})$ for some positive M. I would like to deduce that $x‘(t)>0$ as $t\to\infty$.

Answer 1

Why are you comparing the lower limits with the Big O notation? It is highly incorrect. It is used only for upper limits (bounds). As for ypur question, the Big O notation is defined for positive values only, and it will always be positive.