4.4.E: Problems on Limits and Operations in $E^{*}$

Exercise $\PageIndex{1}$

Show by examples that all expressions $\left(1^{*}\right)$ are indeterminate.

Exercise $\PageIndex{2}$

Give explicit definitions for the following "unsigned infinity" limit statements:
$$\text { (a) } \lim _{x \rightarrow p} f(x)=\infty ; \quad \text { (b) } \lim _{x \rightarrow p^{+}} f(x)=\infty ; \quad(\mathrm{c}) \lim _{x \rightarrow \infty} f(x)=\infty .$$

Exercise $\PageIndex{3}$

Prove at least some of Theorems $1-10$ and formulas $(\mathrm{i})-(\mathrm{iv})$ in Note 1.

Exercise $\PageIndex{4}$

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 $x$.]

Exercise $\PageIndex{5}$

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) $n=m(n, m \in N)$.

Exercise $\PageIndex{6}$

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.]

Exercise $\PageIndex{7}$

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 $x=-y=+\infty,$ $z=-1 .$