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 $G(x) \sim  e^x$.