The image shows a differential equation problem. It states that $y(t)$ is a solution to the equation:

$\frac{dy}{dt} = \frac{1}{6} y \left( 1 - \frac{y}{6} \right),$

with the initial condition $y(0) = 12$. The task is to determine:

$\lim_{t \to \infty} y(t).$

To solve this, we can analyze the behavior of the differential equation as $t \to \infty$. Notice that this is a logistic equation, where the term $\left( 1 - \frac{y}{6} \right)$ suggests that the growth of $y(t)$ will slow down as $y(t)$ approaches 6. The system has a carrying capacity at $y = 6$, which is likely where the solution will stabilize as time goes to infinity.

Approach:

To find the limit as $t \to \infty$, consider the behavior of the differential equation when $y(t)$ becomes constant, i.e., when $\frac{dy}{dt} = 0$. Setting the right-hand side of the differential equation to zero:

$0 = \frac{1}{6} y \left( 1 - \frac{y}{6} \right).$

This gives the equation:

$y \left( 1 - \frac{y}{6} \right) = 0.$

From this, we get two possible solutions: $y = 0$ or $y = 6$.

Since the initial condition is $y(0) = 12$, the solution is likely to approach $y = 6$ as time progresses. Therefore:

$\lim_{t \to \infty} y(t) = 6.$

Let me know if you'd like further details on the solution process or additional clarification! Here's a list of related questions:

1. What is the general solution to the logistic differential equation?
2. How does the value of the carrying capacity $K$ affect the long-term behavior of the solution?
3. What is the method to solve separable differential equations like this one?
4. How does the initial condition $y(0) = 12$ influence the behavior of the solution?
5. What happens if the initial condition is less than 6?

Tip: The logistic equation is often used to model population growth, where the growth rate decreases as the population approaches the carrying capacity.