= Solution
The two event types are disjoint and exhaustive among first events, so their <cumulative incidence functions> sum to the <probability> of any event by $t$. Equivalently their derivatives add to $S(t)(h_A(t)+h_B(t))=-S'(t)$. Therefore
$$
\boxed{F_A(t)+F_B(t)=1-S(t)=1-e^{-\theta_A t-\theta_B\min(t,\tau)}.}
$$
For the usual positive disease-death intensity $\theta_A>0$, the limit is one: eventually every individual has an event, even though treatment-side-effect events cease after $\tau$. If the permitted boundary $\theta_A=0$ is included, however,
$$
\boxed{\lim_{t\to\infty}(F_A+F_B)=1-e^{-\theta_B\tau}.}
$$
Then a fraction $e^{-\theta_B\tau}$ remains event-free forever, because all hazards vanish after $\tau$. This includes the no-event case $\theta_A=\theta_B=0$. Thus an assertion that the limit always equals one requires $\theta_A>0$.
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