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Chapter 5: Q.5TF (page 443)

What happens to the function \(P(t)=\frac{100Ae^{100t}}{1+Ae^{100t}}\) as \(t\rightarrow \infty?\) What does that mean about this particular population?

Short Answer

Expert verified

100

Step by step solution

01

Given Information

Consider the function \(P(t)=\frac{100Ae^{100t}}{1+Ae^{100t}}\).

02

Calculate the limit

\(\begin{align*}L&=\lim_{t\rightarrow \infty}P(t)\\&=\lim_{t\rightarrow \infty}\frac{100Ae^{100t}}{1+Ae^{100t}}\\&=\lim_{t\rightarrow \infty}\frac{100A}{e^{-100t}+A}\\&=\frac{100A}{A}\\&=100\end{align*}\)

03

Write the reason

In long term the population is constant.

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