Assume $f(x)$ is a function of $x$. How $f(x)$ should be (orderwise, with respect to $x$) in order to make the following term tend to infinity as $x\to\infty$? $$g(x)=\sqrt{f(x)} \large \exp\left(a x\left(1-\frac{b}{b+\ln(1+c/f(x))}\right)\right)$$
where $a,b,c>0$. In other words, what are the necessary and sufficient conditions on $f(x)$ such that the above term tends to $\infty$ as $x\to\infty$?
My solution:
If $f(x)=o(1)$. then, I can probably say that $g(x)=\sqrt{f(x)} \large \exp\left(a x\right)$ (I am not sure if this step is 100% true, I know I should use taylor expansions, etc to make sure it is right). Since $g(x)=\omega(1), {f(x)}=\omega(\exp(-2a x))$. Therefore, one set of solution is $f(x)=o(1)$ and $f(x)=\omega(\exp(-2a x))$.
Any idea how to solve this?