Yes, I believe the Im(z) remains finite. I'm basically trying to understand how Wikipedia found the limit on this page: Stiff equation - Wikipedia, the free encyclopedia , section "Example: The Euler and trapezoidal methods".
So, about what you did: I assume is the imaginary part of z? Where did the vanish to?
But from what you did, it looks like the limit becomes: (since the other terms vanish).
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Which is what wikipedia got, but is my working right?
it's a basic algebra thing: if we have a quotient Q/P, where Q, P are polynomials on some unknow t of the same degree (on t), then we can always write Q/P = 1 - Q'/P, for some pol. in t of degree at most deg(Q)
After that all is simple limits: as it is x --> -oo all the rest is taken as constants, and thus (x + constant)/(-x + constant) --> -1
Tonio