I keep seeing a property but I not sure what it is called or how exactly it works. Each time I see it used it is with a, say $G()$, whose derivative is bounded, and an input $x + x_n$ where $x_n \approx O(n^{-1})$. Then I see that because of the boundedness (I think)
\begin{align*} G(x + x_n) &\approx G\left (x + O(n^{-1})\right )\\ &\approx G(x) + O(n^{-1}) \end{align*}
Is there something obvious that I am missing?
From Lagrange mean value theorem we have $G(x + x_n) - G(x) = (x + x_n - x) \cdot G'(\xi)$, where $\xi$ is some point in $(x, x + x_n)$. Given that $G'$ is bounded, we have $|G'(\xi)| < M$ for some $M$ and all $\xi$, so $|G(x + x_n) - G(x)| < M \cdot x_n = O(x_n)$, or, equivalently, $G(x + x_n) = G(x) + O(x_n)$. If $x_n = O(n^{-1})$, this becomes $G(x + x_n) = G(x) + O(n^{-1})$.