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The growth rate of $\left(1+\frac{x}{f(x)}\right)^{f(x)}$

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 no …