When does $f(x) \sim g(x) + h(x)$ imply $g(x) \sim f(x) - h(x)$?

Question

When does $f(x) \sim g(x) + h(x)$ imply $g(x) \sim f(x) - h(x)$? I tried to calculate: $$\lim _{x\to \infty }\frac{f-h}{g}$$ But was stuck at: $$ \lim _{x\to \infty }\frac{\frac{g}{g+h}}{\frac{f}{g+h}-1+\frac{g}{g+h}} $$

Answer 1

$$\frac{f-h}{g}=\frac{1-h/f}{\frac{g+h}{f}-h/f}$$

Hence, as $\frac{g+h}{f} \to 1$, a sufficient condition for that to tend to $1$ is that $h/f \to c \ne 1$. Or equivalently, $g/f \to \ell \ne 0$.

Answer 2

Set $u = \dfrac{f}{g+h}$. Then we have $f=u\cdot (g+h)$ and $u\to 1$.

Then $\dfrac{f-h}{g} = \dfrac{u\cdot(g+h)-h}{g}=u+\dfrac{h}{g}(u-1)$.

If $\dfrac{h}{g}$ is bounded then $u+\dfrac{h}{g}(u-1) \to 1$