Problem 1
Show by examples that all expressions \(\left(1^{*}\right)\) are indeterminate.
Problem 2
Give explicit definitions for the following "unsigned infinity" limit statements: (a) \(\lim _{x \rightarrow p} f(x)=\infty\) (b) \(\lim _{x \rightarrow p^{+}} f(x)=\infty\); (c) \(\lim _{x \rightarrow \infty} f(x)=\infty\)
Problem 2
Verify that any infinite set in a discrete space is closed and bounded but not compact. [Hint: In such a space no sequence of distinct terms clusters.]
Problem 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. (a) \(\lim _{x \rightarrow \infty} \frac{1}{x}(=0)\). (b) \(\lim _{x \rightarrow \infty} \frac{x(x-1)}{1-3 x^{2}}\). (c) \(\lim _{x \rightarrow 2^{+}} \frac{x^{2}-2 x+1}{x^{2}-3 x+2}\). (d) \(\lim _{x \rightarrow 2^{-}} \frac{x^{2}-2 x+1}{x^{2}-3 x+2}\). (e) \(\lim _{x \rightarrow 2} \frac{x^{2}-2 x+1}{x^{2}-3 x+2}(=\infty)\).
Problem 5
Prove that if \(f\) is monotone on \((a, b) \subseteq E^{*},\) it has at most countably many discontinuities in \((a, b)\).
Problem 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 (i) \(n>m ;\) (ii) \(n
Problem 5
Show that each arc is a continuous image of [0,1] . [Hint: First, show that any \([a, b] \subseteq E^{1}\) is such an image. Then use a suitable composite mapping.]
Problem 6
Show that every polynomial of degree one on \(E^{n}\left({ }^{*}\right.\) or \(\left.C^{n}\right)\) is uniformly continuous.
Problem 7
Prove that \(A\) is connected iff there is no continuous map $$ f: A \underset{\text { onto }}{\longrightarrow}\\{0,1\\} .^{5} $$ [Hint: If there is such a map, Theorem 1 shows that \(A\) is disconnected. (Why?) Conversely, if \(A=P \cup Q(P, Q\) as in Definition 3), put \(f=0\) on \(P\) and \(f=1\) on \(Q\). Use again Theorem 1 to show that \(f\) so defined is continuous on \(A\).]
Problem 7
Find (if possible) the ordinary, the double, and the iterated limits of \(f\) at (0,0) assuming that \(f(x, y)\) is given by one of the expressions below, and \(f\) is defined at those points of \(E^{2}\) where the expression has sense. (i) \(\frac{x^{2}}{x^{2}+y^{2}} ;\) (ii) \(\frac{y \sin x y}{x^{2}+y^{2}} ;\) (iii) \(\frac{x+2 y}{x-y}\); (iv) \(\frac{x^{3} y}{x^{6}+y^{2}}\) (v) \(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\) (vi) \(\frac{x^{5}+y^{4}}{\left(x^{2}+y^{2}\right)^{2}} ;\) (vii) \(\frac{y+x \cdot 2^{-y^{2}}}{4+x^{2}}\) (viii) \(\frac{\sin x y}{\sin x \cdot \sin y}\).
