If $g(x) \succ f(x)$ (or $\lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}=0$), will $g(x)-f(x)$ always be a strictly increasing function?
2026-04-04 03:23:12.1775272992
Monotonicity of $f(x)-g(x)$ where $g$ is asymptotically greater than $f$
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No, it can oscillate, consider $f(x) \equiv 1$ and $g(x) = e^x+ \sqrt{x}\sin e^{x^2}$. On average, the difference grows, but it oscillates while doing so.