4.4.E: Problems on Limits and Operations in \(E^{*}\)
- Page ID
- 23721
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Show by examples that all expressions \(\left(1^{*}\right)\) are indeterminate.
Give explicit definitions for the following "unsigned infinity" limit statements:
\[
\text { (a) } \lim _{x \rightarrow p} f(x)=\infty ; \quad \text { (b) } \lim _{x \rightarrow p^{+}} f(x)=\infty ; \quad(\mathrm{c}) \lim _{x \rightarrow \infty} f(x)=\infty .
\]
Prove at least some of Theorems \(1-10\) and formulas \((\mathrm{i})-(\mathrm{iv})\) in Note 1.
In the following cases, find \(\lim f(x)\) in two ways: (i) use definitions only; (ii) use suitable theorems and justify each step accordingly.
\[
\begin{array}{l}{\text { (a) } \lim _{x \rightarrow \infty} \frac{1}{x}(=0) . \quad \text { (b) } \lim _{x \rightarrow \infty} \frac{x(x-1)}{1-3 x^{2}}} \\ {\text { (c) } \lim _{x \rightarrow 2^{+}} \frac{x^{2}-2 x+1}{x^{2}-3 x+2} \text { (d) } \lim _{x \rightarrow 2^{-}} \frac{x^{2}-2 x+1}{x^{2}-3 x+2}} \\ {\text { (e) } \lim _{x \rightarrow 2} \frac{x^{2}-2 x+1}{x^{2}-3 x+2}(=\infty)}\end{array}
\]
[Hint: Before using theorems, reduce by a suitable power of \(x\).]
Let
\[
f(x)=\sum_{k=0}^{n} a_{k} x^{k} \text { and } g(x)=\sum_{k=0}^{m} b_{k} x^{k}\left(a_{n} \neq 0, b_{m} \neq 0\right) .
\]
Find \(\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}\) if \((\mathrm{i}) n>m ;(\text { ii }) n<m ;\) and (iii) \(n=m(n, m \in N)\).
Verify commutativity and associativity of addition and multiplication in \(E^{*},\) treating Theorems \(1-16\) and formulas \(\left(2^{*}\right)\) as definitions. Show by examples that associativity and commutativity (for three terms or more) would fail if, instead of \(\left(2^{*}\right),\) the formula \(( \pm \infty)+(\mp \infty)=0\) were adopted.
[Hint: For sums, first suppose that one of the terms in a sum is \(+\infty ;\) then the sum is + \(\infty\). For products, single out the case where one of the factors is \(0 ;\) then consider the infinite cases.]
Continuing Problem \(6,\) verify the distributive law \((x+y) z=x z+y z\) in \(E^{*},\) assuming that \(x\) and \(y\) have the same sign (if infinite), or that \(z \geq 0\). Show by examples that it may fail in other cases; e.g., if \(x=-y=+\infty,\) \(z=-1 .\)