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