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 $p\left(t\right)=\frac{1500}{3+2e^{-0.1t}}$.

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1. Identify the function that describes the system: p(t) = \frac{1500}{3 + 2e^{-0.1t}}.
2. To determine if a steady state exists, evaluate the limit of p(t) as t approaches infinity: \lim_{t \to \infty} p(t).
3. Observe that as t approaches infinity, the term e^{-0.1t} approaches zero because the exponential function decays to zero.
4. 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}.
5. Calculate the simplified expression to find the steady-state value: \frac{1500}{3}.