Let's solve the problem step by step: ### 1. **Interpret the Direction Field** The direction field is for an autonomous differential equation: \[ \frac{dy}{dx} = f(y) \] We're told that the solution includes the point \( y(1) = 0 \). ### 2. **Look for Equilibrium Solutions** Equilibrium solutions (constant solutions) occur where the slope is zero everywhere—i.e., where \( f(y) = 0 \). From the graph, the horizontal direction segments are at \( y = 0 \) and \( y = -2 \). This means at these values, \( f(y) = 0 \), so these are equilibrium solutions. ### 3. **Analyze the Behavior Around Equilibria** - For \( y > 0 \), the direction field arrows tend to point downwards (decreasing). - For \( -2 < y < 0 \), the arrows are pointing upwards (increasing). - For \( y < -2 \), the arrows are pointing downwards (decreasing). ### 4. **Find the Long-term Behavior** Since the solution starts at \( y(1) = 0 \), and \( y = 0 \) is an equilibrium, let's check if the solution stays at \( y = 0 \): - At \( y = 0 \), the slope is zero, so if you start at \( y = 0 \), you stay at \( y = 0 \). - Thus, as \( t \to \infty \), the solution remains at \( y = 0 \). ### 5. **Conclusion** \[ \lim_{t \to \infty} y(t) = 0 \] ### **Final Answer** \[ \boxed{0} \]