4.4.E: Problems on Limits and Operations in \(E^{*}\)
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 \(x=-y=+\infty,\) \(z=-1 .\)