If a function f represents a system that varies in time, the existence of lim ${\displaystyle\lim_{t\rightarrow\infty}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a colony of squirrels is given by $p (t) = \frac{1500}{3 + 2 e^{- 0.1 t}}$ .

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Identify the function that describes the system: p(t) = \frac{1500}{3 + 2e^{-0.1t}}.

To determine if a steady state exists, evaluate the limit of p(t) as t approaches infinity: \lim_{t \to \infty} p(t).

Observe that as t approaches infinity, the term e^{-0.1t} approaches zero because the exponential function decays to zero.

Substitute e^{-0.1t} with 0 in the expression for p(t) to simplify the limit: \lim_{t \to \infty} \frac{1500}{3 + 2 \cdot 0}.

Calculate the simplified expression to find the steady-state value: \frac{1500}{3}.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit of a Function

The limit of a function describes the behavior of that function as the input approaches a certain value, which can be finite or infinite. In this context, the limit as t approaches infinity indicates how the function behaves as time progresses indefinitely. Understanding limits is crucial for analyzing the long-term behavior of dynamic systems, such as population models.

Steady State (Equilibrium)

A steady state, or equilibrium, occurs when a system's variables no longer change over time, meaning that the system has reached a stable condition. In mathematical terms, this is often represented by the limit of a function as time approaches infinity being a constant value. Identifying steady states is essential in various fields, including biology and economics, to predict long-term outcomes.

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value, often modeled by functions involving the exponential function e. In the given population model, the term e^{-0.1t} represents the decay of the influence of initial conditions over time, which is critical for determining how the population stabilizes as time progresses.

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