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 culture of tumor cells is given by $p\left(t\right)=\frac{3500t}{t+1}$.

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1. First, identify the function given: \( p(t) = \frac{3500t}{t+1} \). This function represents the population of tumor cells over time.
2. To determine if a steady state exists, we need to evaluate the limit of \( p(t) \) as \( t \) approaches infinity: \( \lim_{t \to \infty} \frac{3500t}{t+1} \).
3. Simplify the expression by dividing the numerator and the denominator by \( t \), the highest power of \( t \) in the denominator: \( \frac{3500t/t}{(t+1)/t} = \frac{3500}{1 + 1/t} \).
4. As \( t \to \infty \), the term \( 1/t \) approaches 0. Therefore, the expression simplifies to \( \frac{3500}{1 + 0} = 3500 \).
5. Conclude that the limit exists and the steady-state value of the population is 3500. This means the population of tumor cells approaches 3500 as time goes to infinity, indicating a steady state.

Key Concepts

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 populations or physical processes.

Steady State (Equilibrium)

A steady state, or equilibrium, occurs when a system's variables remain constant over time, indicating that the system has reached a balance. In mathematical terms, this is often represented by the limit of a function equating to a constant value as time approaches infinity. Identifying steady states is essential in various fields, including biology and physics, to predict system behavior under stable conditions.

Rational Functions

A rational function is a ratio of two polynomial functions. In the given example, the population function p(t) = 3500t / (t + 1) is a rational function where the numerator and denominator are both polynomials. Analyzing rational functions involves understanding their limits, asymptotic behavior, and potential steady states, which are critical for determining the long-term behavior of the system they represent.