If a function f represents a system that varies in time, the existence of lim $\underset{t\to \infty }{\lim }f\left(t\right)$ 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 bacteria culture is given by $p\left(t\right)=\frac{2500}{t+1}$.

Verified step by step guidance

1. Identify the function given: \( p(t) = \frac{2500}{t+1} \).
2. Understand that we need to find \( \lim_{t \to \infty} p(t) \) to determine if a steady state exists.
3. Set up the limit: \( \lim_{t \to \infty} \frac{2500}{t+1} \).
4. Recognize that as \( t \to \infty \), the denominator \( t+1 \) becomes very large, making the fraction approach zero.
5. Conclude that the limit is zero, indicating the system reaches a steady state at a population of 0.

Key Concepts

Limits

In calculus, a limit describes the behavior of a function as its input approaches a certain value. Specifically, the limit as t approaches infinity examines how the function behaves as time progresses indefinitely. Understanding limits is crucial for determining the long-term behavior of dynamic systems, such as whether they stabilize or diverge.

Steady State

A steady state in a system occurs when the variables of interest no longer change over time, indicating that the system has reached equilibrium. Mathematically, this is represented by the existence of a limit as time approaches infinity. In the context of the given function, identifying the steady-state value involves evaluating the limit of the population function as time increases.

Population Dynamics

Population dynamics is a field of study that examines how populations change over time due to various factors such as birth, death, and resource availability. The function provided, p(t) = 2500/(t+1), models the growth of a bacterial culture, illustrating how the population evolves and approaches a maximum capacity as time progresses. Understanding these dynamics is essential for analyzing the behavior of biological systems.