In my Numerical Methods for PDEs textbook by Ari Uscher L-stability and stiff decay are introduced by considering a generalized test equation:
$y' = \lambda (y - g(t)), 0 < t < b$
where g(t) is a bounded but otherwise arbitrary function. As $Re(\lambda) \rightarrow -\infty$ the exact solution satisfies $y(t) \rightarrow g(t), 0< t < b$. My first question (1) is that this is not obvious to me and no explanation is given. How can one conclude that?
It goes on to define stiff decar for a method as $|y_n - g(t_n)| \rightarrow 0$ as $\Delta t Re(\lambda) \rightarrow 0$. My second question (2) is how this is found in practice. If I want to determine if an ODE is stiff, how do I find/choose a g(t) to satisfy the above requirement? Or does any arbitrary bounded g satisfy it if my ODE is stiff as the test equation supposedly shows (which I don't understand and is listed as question 1).