Let t \in \mathbb{R} be large.
Let f be a function over [0,t] satisfying f(0) = 1 and f'(x) = e^{-f(x)} for all x.
Let g be a function over [0,t] satisfying g(0) = 1 and g'(x) = (1 - g(x)/t^2)^{t^2} for all x. Note that g'(x) \sim e^{-g(x)}.
Without solving the two...