91Ó°ÊÓ

Exponential growth generally occurs when population growth is density independent. List conditions under which a population might stop growing exponentially.

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{n+1}{n^{2}-1}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{n+2}{n^{2}-4}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left[\left(\frac{1}{3}\right)^{n}+\left(\frac{1}{2}\right)^{n}\right] $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(3^{-n}-4^{-n}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty} \frac{n+2^{-n}}{n} $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty} \frac{n+3^{-n}}{n} $$

In Problems \(93-102\), the sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{1}{2} a_{n}+2 $$

The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{1}{3} a_{n}+\frac{4}{3} $$

The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{2}{5} a_{n}-\frac{9}{5} $$