Textbook Question
Use the precise definition of infinite limits to prove the following limits.
1. Limits and Continuity
Introduction to Limits
2:20 minutesProblem 4.3.111a
Textbook Question
Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.
a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?
Verified step by step guidance1
First, understand the concept of carrying capacity in population models. It refers to the maximum population size that the environment can sustain indefinitely.
To find the carrying capacity, we need to evaluate the limit of the function P(t) as t approaches infinity. This involves analyzing the behavior of the function as t becomes very large.
The function given is P(t) = \( \frac{Kt^2}{t^2 + b} \). Substitute K = 300 and b = 30 into the function, resulting in P(t) = \( \frac{300t^2}{t^2 + 30} \).
To find \( \lim_{t \to \infty} P(t) \), divide both the numerator and the denominator by \( t^2 \), the highest power of t in the expression. This simplifies the function to \( \frac{300}{1 + \frac{30}{t^2}} \).
As t approaches infinity, \( \frac{30}{t^2} \) approaches 0, simplifying the expression to \( \frac{300}{1} \). Therefore, the limit is 300, which is the carrying capacity of the population.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
In calculus, a limit describes the behavior of a function as its input approaches a certain value. It is essential for understanding how functions behave at boundaries or at infinity. In this context, evaluating the limit as t approaches infinity helps determine the long-term behavior of the population model.
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Carrying Capacity
Carrying capacity refers to the maximum population size that an environment can sustain indefinitely without being degraded. In population models, it is often represented as the limit of the population function as time approaches infinity, indicating the population's equilibrium state under given conditions.
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Rational Functions
A rational function is a ratio of two polynomials. In the given population model, P(t) is a rational function where the numerator and denominator are both polynomials in t. Understanding the behavior of rational functions, particularly their limits at infinity, is crucial for analyzing population dynamics in this context.
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