I am reading this ML paper on Influence Functions and have a problem understanding Appendix A, equation (13). Omitting all irrelevant context from the paper, imagine that we have a function
$$f(\epsilon) = (A + \epsilon B)^{-1} \cdot \epsilon v$$
where all encountered matrices are nice and invertible, $v$ is some vector. Authors claim in the equation 13, that as $\epsilon \to 0$ we got:
$$f(\epsilon) = \epsilon Av + o(\epsilon)$$
In other words, we can discard $\epsilon B$ as being $O(\epsilon^2)$. But how is it so? By using the identity $\frac{1}{\lambda} M^{-1} = (\lambda M)^{-1}$, we can see, that:
$$ f(\epsilon) = (A + \epsilon B)^{-1} \cdot \epsilon v = (\frac{1}{\epsilon} A + B)^{-1} \cdot v $$
So there is no $\epsilon$ term in front of the matrix $B$ whatsoever! How can we discard it as being $o(\epsilon)$?
$(A+\epsilon B)^{-1} = \big(A(I+\epsilon A^{-1}B)\big)^{-1}= (I+\epsilon A^{-1}B)^{-1}A^{-1}=\big(I-\epsilon A^{-1}B + O(\epsilon^2)\big)A^{-1}$. Now multiply by $\epsilon v$.