A problem which circulated among the grad students here at Penn today, which came up while somebody was teaching differential equations:
Consider the differential equation dy/dx = (y-1)2. Separate variables and integrate both sides as usual; you get y(x) = 1 - 1/(x+C), where C is determined by the initial condition. Take the limit as x goes to infinity, and you get limx -> ∞ y(x) = 1, regardless of C.
But now notice that dy/dx is positive for all y. (We're working over the reals here.) So if the initial condition is of the form y(x0) = y0 for some y0 > 1, then we start at 1 and keep going upwards; how can the limit be 1?
Of cousre there's a mistake somewhere in here. But where is it? (I know the answer, but it took annoyingly long to figure out.)
edit: the differential equation was wrong.