I was reading this question on functions "in the middle" of linear and exponential growth, and it caused me to think of this function:

$$ G(x) = \left(1 + \frac{x}{f(x)}\right)^{f(x)} $$

for some non-decreasing function $f$. If $f(x)$ is constant, then $G$ is just a polynomial in $x$. If $f(x) \geq x^2$, then I believe $G(x) \to e^x$ as $x \to \infty$ (edit: should be $G(x) \sim e^x$). This might be true for other $f(x) = \omega(x)$, I'm not sure.

Three questions:

- Is $G$ of interest or use anywhere, or has its growth rate been studied? Is there an interesting connection I'm missing?
- What can we say about how fast $G$ grows for choices of $f$ between constant and $x^2$? In particular, can we say something interesting or simpler about how fast these grow?
- $\left(1 + \sqrt{x}\right)^{\sqrt{x}}$
- $\left(1 + x^{\alpha}\right)^{x^{1-\alpha}}$
- $\left(1+\frac{x}{\log x}\right)^{\log x}$
- What I was sort of hoping is that a choice of $f$ would give a function so that $G(G(x)) = \Theta(e^x)$; does that seem possible?