In sec.50 of this book, Landau discussed penetration through a barrier, and in the note he mentioned
In a passage from right to left through the lower half-plane, the function $\psi(x)$ at first increases and then decreases in modulus, becoming an exponentially small quantity on the left-hand axis ($\phi\to-\pi$), which it would not be legitimate to keep superimposed on the exponentially large function (50.2). In the region where $\psi(x)$ is exponentially large, the inexactness of the quasi-classical approximation loses the exponentially small correction which for $\phi\to-\pi$ could become an exponentially large term, and the latter is therefore lost also.
I don't really understand the last sentence. How does the dropped exponentially small term become exponentially large on the left-hand axis?
