Week 2: May 19 $^$th - May 25 $^$th

Suggested Problems

Problems you may find instructive, or that I find interesting.

2.1

#27, 30, 32, 40, 45, 46 & 47

(Beware that #49 is misprinted.)

2.2

#35, 39, 41, 46, 51, 52, 54 & 61(B)

2.3

#20, 25, 29, 30, 43, 44, 45, 51, 57, 60 & 61

(#56 is also fun, if you have time. It is a bit like Russell's Paradox.)

2.5

#5, 7, 13, 17, 31, 33, 43, 45, & 50

(If you want more practice with $\epsilon-\delta$ , it is possible to prove #50 without the Squeeze Theorem.)

Assigned Problems

Due May 26 $^$th , in lecture.

1. Prove that 2.2.6(ii) implies 2.2.6(i); that is,
      ``$$\lim_{h\to 0} f(c+h) = L$$   ''$$\Longrightarrow$$   ``$$\lim_{x\to c}f(x) = L$$   ''$$.$$
   For clarity, use the notation $\epsilon$ , $\delta_{(ii)}$ and $\delta_{(i)}$ .

2. Give an $\epsilon-\delta$ proof that,
   $$\lim_{x\to 1^+}\frac{1}{x^2+2} = \frac{1}{3}.$$
   Hint: to find $x-1$ , first write $\left\lvert \frac{1}{x^2+2}-\frac{1}{3} \right\rvert$ as a single fraction and factor the numerator.

3. Function $f(x)$ with domain $(-\infty,\infty)$ is Lipschitz if
   $$\begin{aligned} \left\lvert f(a)-f(b) \right\rvert \leq \left\l... ...ert &&\text{ for any two values } a \text{ and } b. \end{aligned}$$
   Prove that $\lim_{x\to c}f(x) = f(c)$ for every number $c$ .