I was reading [this question](https://mathoverflow.net/questions/4347/ffx-expx-1-and-other-functions-just-in-the-middle-between-linear-and-exp) 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?



The growth rate of $\left(1+\frac{x}{f(x)}\right)^{f(x)}$
Let's suppose $x/f(x) \to 0$. Then take logarithm. As $x \to \infty$,
$$
\log G(x) = f(x) \log\left(1+\frac{x}{f(x)}\right)
=f(x)\left(\frac{x}{f(x)}+O\left(\left(\frac{x}{f(x)}\right)^2\right)\right)
\\
= x + O\left(\frac{x^2}{f(x)}\right)
\tag{\*}$$
Now if we have the stronger $x^2/f(x) \to 0$, then
$$
G(x) = \exp\left(x + O\left(\frac{x^2}{f(x)}\right)\right)
=e^x\exp \left(O\left(\frac{x^2}{f(x)}\right)\right)
=e^x\left(1+o(1)\right)
\tag{\*\*}$$
But note: it is incorrect to write this as $G(x) \to e^x$. Instead write $G(x) \sim e^x$. 


Now consider the three examples. See if I made any mistakes.  

..........  

$$
G\_1(x) = (1+\sqrt{x}\;)^{\sqrt{x}}
\\
\log G\_1(x) = \sqrt{x}\log(1+\sqrt{x}\;) = \sqrt{x}\log\left(\sqrt{x}\left(1+\frac{1}{\sqrt{x}}\right)\right)
\\
=\sqrt{x}\left(\log\sqrt{x}+\log\left(1+\frac{1}{\sqrt{x}}\right)\right)
=\sqrt{x}\left(\frac{1}{2}\log x+\frac{1}{\sqrt{x}}+O\left(\frac{1}{x}\right)\right)
\\
G\_1(x) = e^{(1/2)\sqrt{x}\log x} e^1 (1+o(1))
\\
G\_1(x) \sim e\;x^{(1/2)\sqrt{x}}
$$  

..........  

Assume $\alpha>1/2$  

$$
G\_2(x) = (1+x^\alpha)^{x^{1-\alpha}}
\\
\log G\_2(x) = x^{1-\alpha}\log\left(1+x^\alpha\right)
=x^{1-\alpha}\left(\log x^\alpha+ \log\left(1+x^{-\alpha}\right)\right)
\\
=x^{1-\alpha}\left(\alpha \log x+x^{-\alpha} +O\left(x^{-2\alpha}\right)\right)
\\
G\_2(x) = e^{x^{1-\alpha}\alpha\log x} e^{x^{1-2\alpha}} (1+o(1))
\sim x^{x^{1-\alpha}\alpha}
$$  

..........  

$$
G\_3(x) = \left(1+\frac{x}{\log x}\right)^{\log x}
\\
\log G\_3(x) = \log x \log\left(1+\frac{x}{\log x}\right)
=\log x\left(\log\frac{x}{\log x}+O\left(\frac{\log x}{x}\right)\right)
\\
=(\log x)^2 -\log x \log\log x+o(1)
\\
G\_3(x) \sim e^{(\log x)^2-\log x \log\log x} = x^{\log x - \log\log x}
$$


