Chapter 2

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{n+1}{n}, \epsilon=.05 $$

In Problems , graph the line \(N_{t+1}=R N_{t}\) in the \(N_{t}-N_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0\), 1\. and 2. for the given value of \(N_{0}\). $$ R=\frac{1}{2}, N_{0}=16 $$

Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{n^{2}}{n^{2}+1}, \epsilon=0.01 $$

In Problems , graph the line \(N_{t+1}=R N_{t}\) in the \(N_{t}-N_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0\), 1\. and 2. for the given value of \(N_{0}\). $$ R=\frac{1}{2}, N_{0}=64 $$

In Problems , graph the line \(N_{t+1}=R N_{t}\) in the \(N_{t}-N_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0\), 1\. and 2. for the given value of \(N_{0}\). $$ R=\frac{1}{3}, N_{0}=81 $$

Formal Definition of Limits: In Problems \(65-70\), use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{t \rightarrow \infty} \frac{1}{n}=0 $$

In Problems , graph the line \(N_{t+1}=R N_{t}\) in the \(N_{t}-N_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0\), 1\. and 2. for the given value of \(N_{0}\). $$ R=\frac{1}{4}, N_{0}=16 $$

Formal Definition of Limits: Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{1}{n^{2}}=0 $$

Formal Definition of Limits: Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{1}{n^{2}+1}=0 $$

In Problems , graph the line \(\frac{N_{t}}{N_{t+1}}=\frac{1}{R}\) in the \(N_{t}-\frac{N_{t}}{N_{t+1}}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, \frac{N_{t}}{N_{t+1}}\right), t=0\), 1, 2, for the given value of \(N_{0}\). Find the parent-offspring ratio. $$ R=3, N_{0}=2 $$

Formal Definition of Limits: Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{n+1}{n}=1 $$

Formal Definition of Limits: Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{n}{n+1}=1 $$

In Problems \(71-82\), use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) \(\lim _{n \rightarrow \infty}\left(\frac{1}{n}+\frac{1}{n^{2}}\right)\)

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

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

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

A bird population lives in a habitat where the number of nesting sites is a limiting factor in population growth. In which of the following cases would you expect that the growth of this bird population over the next few generations could be reasonably well approximated by exponential growth? (a) All nesting sites are occupied. (b) The bird population just invaded the habitat, and the population size is still much smaller than the available nesting sites. (c) In the previous year, a hurricane killed more than \(90 \%\) of the birds in this habitat.

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

= Pollen records show that the number of Scotch pine (Pinus sylvestris) grew exponentially for about 500 years after colonization of the Norfolk region of Great Britain about 9500 years ago. Can you find a possible explanation for this growth?

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-5}{n^{2}}\right) $$

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 \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{2}{5} a_{n}-\frac{9}{5} $$

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

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

The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{7}{a_{n}} $$

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

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

The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\sqrt{7 a_{n}} $$

In Problems 103-110, assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed ooints of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which ixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{2}\left(a_{n}+5\right), a_{0}=1 $$

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed ooints of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which ixed point is the limiting value for the given initial condition. $$ a_{n+1}=2 a_{n}\left(1-a_{n}\right), a_{0}=0.1 $$

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed ooints of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which ixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{4}{a_{n}}\right), a_{0}=1 $$