Let $U(x) = \exp(\frac{x}{1-x})$. For what function $f$ do we have $\lim\limits_{t \to 1} \frac{U(t+xf(t))}{U(t)} = e^x\,$ for all $x\;?$

Question

Let $\;U(x)=\exp\left(\dfrac{x}{1-x}\right)\;$ for $\;0\leqslant x<1\,.$

For which positive functions $f$ defined on the interval $(0,1)$ do we have that

$\lim\limits_{t\to1}\dfrac{U(t+xf(t))}{U(t)}=e^x\;$ for all $x\in\mathbb{R}\;?$

For added context, this should be equivalent to proving that the maximum of distribution $\;F(x)=1-\exp\left(\dfrac{-x}{1-x}\right)\;$ for $\,0\leqslant x<1\,$ can be normalized to converge to the Gumbel distribution.

Thank you.