I have the following equation, where $X$ is a matrix, $f$ is a linear function of matrices, and $0<\varepsilon\ll1$:
$$X=\frac{Xf(X)}{\varepsilon}+\tanh\left(\frac{Xf(X)}{\varepsilon}\right)$$
I've tried an asymptotic expansion of the form:
$$X=X_0+\varepsilon X_1+\varepsilon^2 X_2+\cdots$$
but I'm not sure how to expand $\tanh$ in terms of powers of $\varepsilon$. If $X=O(1)$ then $\frac{Xf(X)}{\varepsilon}$ is very large and I'm not sure of the series representation of $\tanh$ can be used.