First-Order Equations · Superchallenge problem

Let $y(t)$ solve $y'=y^2(1-y)$ with $y(0)=\dfrac12$. Then $y(t)\to1$ as $t\to\infty$, and $1-y(t)$ decays like a multiple of $e^{-t}$. Find $\displaystyle\lim_{t\to\infty}e^{t}\bigl(1-y(t)\bigr)$.