Well, what about two functions that have equal asymptotes as they approach infinity?
I've been wondering this for a while. Given two C-infinity functions f(x) and g(x) such that f(x) != g(x), is it possible for f(x) = g(x) for some continuous range (a, b)?
Basically, I first wanted to know if it was possible for any functions, then realized that there are cheap ways out like absolute value functions and defining the function to be different for different ranges. That's why I'm asking specifically about C-infinity functions (all derivatives, and derivatives of derivatives, and so-on are continuous).
Which, by the way, shows that the Taylor's series of a function does NOT necessarily converge to that function! The Taylor's series, about x= 0, for this function. is just "0" and converges to 0 for all x.
Of course, this function is not "analytic" in any neighborhood of 0.