{Use of Tech} Population growth Consider the following population functions.
d. Evaluate and interpret lim t→∞ p(t).
p(t) = 600 (t²+3/t²+9)

Verified step by step guidance

1. Step 1: Identify the function p(t) = 600 \(\left\)(\(\frac{t^2 + 3}{t^2 + 9}\)\(\right\)) and recognize that you need to evaluate the limit as t approaches infinity.
2. Step 2: Simplify the expression \(\frac{t^2 + 3}{t^2 + 9}\) by dividing both the numerator and the denominator by t^2, the highest power of t in the expression.
3. Step 3: After simplification, the expression becomes \(\frac{1 + \frac{3}{t^2}\)}{1 + \(\frac{9}{t^2}\)}.
4. Step 4: Evaluate the limit of the simplified expression as t approaches infinity. As t becomes very large, the terms \(\frac{3}{t^2}\) and \(\frac{9}{t^2}\) approach 0.
5. Step 5: Conclude that the limit of the expression \(\frac{1 + \frac{3}{t^2}\)}{1 + \(\frac{9}{t^2}\)} as t approaches infinity is 1, and therefore, lim_{t \(\to\) \(\infty\)} p(t) = 600 \(\times\) 1 = 600. Interpret this as the population stabilizing at 600 as time goes to infinity.

Key Concepts

Limits

In calculus, a limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value. Evaluating limits helps us understand the function's behavior at points where it may not be explicitly defined or at infinity. In this case, we are interested in the limit of the population function p(t) as t approaches infinity, which provides insight into the long-term behavior of the population.

Population Functions

Population functions model the growth of a population over time, often represented as p(t), where t is time. These functions can take various forms, such as exponential or logistic growth models. Understanding the specific form of the population function given, p(t) = 600(t² + 3)/(t² + 9), is crucial for evaluating its behavior as time progresses, particularly as t approaches infinity.

Asymptotic Behavior

Asymptotic behavior refers to the behavior of a function as its input approaches a limit, often infinity. In the context of population growth, it helps us determine the maximum population size that can be sustained over time. By analyzing the limit of p(t) as t approaches infinity, we can interpret the long-term population trend and understand whether the population stabilizes, grows indefinitely, or declines.