I have a probability generating function
$$ G(z) = \Bigg(\frac{ 1-2d + \sqrt{1-4d(1-d)z}}{2(1-2d)}\Bigg)^{\frac{1-2d}{d^2}\kappa}\ \Bigg(\frac{1-\sqrt{1-4d(1-d)z}}{2dz}\Bigg)^{\frac{\kappa}{d^2}}$$
and I'd like to expand this for $d\rightarrow 0$. This expansion should retain the property $G(1)=1$.
My attempt was to use $$ \sqrt{1-4d(1-d)z} \approx 1-2dz$$ within both numerators of this expression, and this obtains $$G(z) \approx \Bigg(\frac{1-\frac{d}{1-d}z}{1-\frac{d}{1-d}}\Bigg)^{\frac{1-2d}{d^2}\kappa}. $$
However, I do not know if this is valid, since I neglected terms of $O(d^2)$ in the two terms, while I kept terms of $O(1/d^2$) in the exponent...
Is there some better way to do this limit $d\rightarrow 0$?