Limits

Abstract

The set real numbers is defined as the set of infinite decimals. Density of the set of rational numbers and of the set of irrational numbers in the set of real numbers is established. This naturally leads to a discussion of accumulation points that serves as a precursor for the main part of this chapter: the theory of limits of functions. Convergence of sequences and series of numbers are discussed briefly. A bounded function, the Dirichlet function, that does not have a limit at any real number is presented. Section 1.8 contains a proof of Steinhaus’ three distance conjecture.

Appendices

Problems

1.1 Problems for Sect. 1.1

• 1. Show that $1.23\overline {9}=1.24.$

• 2. Carry out the division algorithm for $17/7.$

• 3. Find the infinite decimal form of $1/7.$ Show all the steps needed to perform the long division.

• 4. When calculating the decimal form of $p/q$ there are $q$ possible remainders. Why is the length of the repeating part at most $q-1?$

• 5. Show $23.6\overline {321}$ is rational.

• 6. Find integers $p$ and $q$ such that $3.14\overline {15}=\frac {p}{q}.$

• 7. Prove that any interval contains a rational number.

• 8. Show $\mathbb {Q}+i\mathbb {Q}:=\{a+ib\mid a,b\in \mathbb {Q}\}$ is dense in $\mathbb {C}.$ Hint: A ball contains an open square with the same center.

• 9 Give a detailed proof of the density of irrationals in the case $x<0.$

• 10 Let $\mathbb {F}$ be the set of finite decimals. Show that $\mathbb {F}$ is dense in $\mathbb {R}.$

1.2 Problems for Sect. 1.2.

In the spirit of Exercise 1.2.2, that is, when verifying that a point is an accumulation point, it is not necessary to check all $\varepsilon>0.$ This is explored in the following problem.

• 1. A number $a$ is an accumulation point of the set $D$ iff for all $k\in \mathbb {N},$ there is an $x\in D,$ such that $0<|x-a|<1/10^{k}.$ [Hint for one part: given any $\varepsilon>0,$ there is $k\in \mathbb {N},$ such that $\varepsilon>\frac {1}{10^{k}}.$]

• 2. If $D\subseteq \mathbb {R}$ and Im$(c)\neq 0,$ then $c$ is not an accumulation point of $D.$

• 3. Prove $0$ is the only accumulation point of $D:=\left \{ \frac {1}{n}\mid n\in \mathbb {N}\right \} .$

• 4. If $c$ is an accumulation point of $D$ and $r>0,$ then $c$ is an accumulation point of $D\cap B_{r}(c).$

• 5. Any real number is an accumulation point of $\mathbb {Q}.$

• 6. Show the imaginary unit $i$ is not an accumulation point of the set of real numbers $\mathbb {R}.$

• 7. Show that if $B_{r}(z)\subseteq A\subseteq \overline {B}_{r}(z),$ then the set of accumulation points of $a$ equals $\overline {B}_{r}(z).$

• 8. Let $\mathbb {F}$ be the set of finite decimals. Show that every real number is an accumulation point of $\mathbb {F}.$

1.3 Problems for Sect. 1.3

• 1. Show $|x|\to 0$ as $x\to 0.$

• 2. Let $f:[0,\infty [\to \mathbb {R}$ be determined by $f(x):=\sqrt {x+3}$. Prove $f(x)\to 2$ as $x\to 1.$

  The following two problems are about limits of the function $f:[0,\infty [\to \mathbb {R}$ determined by $f(x):=\sqrt {x}.$ Since $D=[0,\infty [,$ only $x\geq 0$ play a role in Eq. (1.4).

• 3. Show $f(x)\to 2$ as $x\to 4.$

• 4. Show $f(x)\to 0$ as $x\to 0.$

• 5. If $f:\mathbb {R}\to \mathbb {R}$ is determined by $f(x)=x-x^{2}$ and $a=\frac {1}{2}.$ Then $f(a)$ $=f\left (\frac {1}{2}\right )$ $=\frac {1}{4}>0.$ Find $\delta>0$ satisfying the conclusions of Theorem 1.3.14..

• 6. Let $f:\mathbb {R}\to \mathbb {R}$ be determined by $f(x):=4-x^{2}.$ Then $f(0)=4.$ Find a $\delta>0,$ such that

  $$ 0<\left|x-0\right|<\delta\implies2<f(x). $$

• 7. If $\left |f(x)\right |\leq 7$ for all $x$ and $g(x):=\left (x-3\right )f(x)$ prove

  $$ g(x)\to0\text{ as }x\to3. $$

• 8. Let $f:\mathbb {R}\to \mathbb {R}$ be determined by

  $$ f(x):=\begin{cases} x-3 & \text{when }x\text{ is rational}\\ 5-x & \text{when }x\text{ is irrational} \end{cases}. $$

  If $x_{0}\neq 4,$ prove $\lim _{x\to x_{0}}f(x)$ does not exist.

• 9. If $\lim _{x\to a}f(x)$ exists and for any $\delta>0,$ there is an $x\in D$ with $0<|x-a|<\delta ,$ such that $f(x)\geq 0,$ then $\lim _{x\to a}f(x)\geq 0.$

  More generally, if $f(x)\to L$ as $x\to a$ and $M\neq L,$ then $f$ is not close to $M$ when $x$ is close to $a.$ This is the content of the following problem.

• 10. Let $f:D\to \mathbb {C}.$ If $f(x)\to L$ as $x\to a$ and $M\neq L,$ then

  $$ \exists\delta>0,\forall x\in D,0<|x-a|<\delta\implies|f(x)-M|>\frac{1}{2}|L-M|. $$

• 11. Prove $x^{2}\to -1$ as $x\to i.$

• 12. Let $D$ be a subset of $\mathbb {C}.$ Let $f,g:D\to \mathbb {C}.$ Suppose $0$ is an accumulation point of $D.$ If $|g(x|\leq M$ and $f(x)=xg(x)$ for all $x\in D,$ then $f(x)\to 0$ as $x\to 0.$

• 13. Let $f,g:D\to \mathbb {C}.$ Suppose $a$ is an accumulation point of $D.$ If $f(x)\to 0$ as $x\to a$ and $|g(x)|\leq 47$ for all $x\in D,$ then $f(x)g(x)\to 0$ as $x\to a.$

• 14. Let $f:D\to \mathbb {C}.$ Suppose $a$ is an accumulation point of $D.$ If $f(x)\to L$ as $x\to a,$ then Re $\left (f(x)\right )\to {\text Re} \left (L\right )$ as $x\to a.$

• 15. Let $f:D\to \mathbb {C}.$ Suppose $x_{0}$ is an accumulation point of $D.$ If ${\text Re} \left (f(x)\right )\to a$ and ${\text Im} \left (f(x)\right )\to b$ as $x\to x_{0},$ then $f(x)\to a+ib$ as $x\to x_{0}.$

• 16. Prove or disprove: If $\lim _{x\to a}f(x)$ exists and $f(x)>0$ for all $x\neq a,$ then $\lim _{x\to a}f(x)>0.$

1.4 Problems for Sect. 1.4

• 1. If $a$ is a finite subset of $\mathbb {C},$ prove that any point in $a$ is an accumulation point of $\mathbb {C}\setminus A.$

• 2. What is wrong with the Composition Rule proposed below?

  “If $f:A\to B,$ $g:B\to C,$ $a$ is an accumulation point of $A,$ $b$ is an accumulation point of $B,$ $f(x)\to b$ as $x\to a,$ and $g(x)\to c$ as $x\to b,$ then $g\circ f(x)\to c$ as $x\to a.$”

• 3. Find two functions $f.g:\mathbb {R}\to \mathbb {R}$ such that neither $\lim _{x\to 0}f(x)$ nor $\lim _{x\to 0}g(x)$ exists, yet both $\lim _{x\to 0}f((x)+g(x)$ and $\lim _{x\to 0}f(x)g(x)$ exist.

• 4. Let $\sigma$ be the pseudo-sine function. Let $f(x):=x\sigma (1/x),$ when $x\neq 0.$ Prove $f(x)\to 0$ as $x\to 0.$

  In the following two problems we assume $x^{1/n}$ exists for all $x\geq 0.$

• 5. Let $D:=[0,\infty [.$ If $a>0$ and $N\in \mathbb {N},$ then $x^{1/n}\to a^{1/n}$ as $x\to a.$

• 6. Let $D:=[0,\infty [.$ For any $N\in \mathbb {N},$ $x^{1/n}\to 0$ as $x\to 0.$

• 7. Suppose $\lim _{x\to a}f(x)=b$ and given any $r>0,$ the set $f\left (B'_{r}(a)\right )$ contains both positive and negative real numbers, show that $b=0.$

• 8. Suppose $\lim _{x\to a}f(x)=b$ and there is a $r>0$ such that $f\left (B'_{r}(a)\right )\subseteq [0,\infty [.$ Show that $b\geq 0.$

• 9. Suppose $\lim _{x\to a}f(x)=b$ and there is a $r>0$ such that $f\left (B'_{r}(a)\right )\subseteq ]0,\infty [.$ Must $b>0?$

• 10. Let $D:=\mathbb {R}.$ If $f(x):=\begin {cases} x & \text {when }x\in \mathbb {Q}\\ 1-x & \text {when }x\notin \mathbb {Q} \end {cases},$ find all values of $a\in \mathbb {R},$ such that $\lim _{x\to a}f(x)$ exists.

• 11. Let $D_{1}:=\left \{ \frac {1}{2n}\mid n\in \mathbb {N}\right \}$ and $D_{2}:=\left \{ \frac {1}{2n-1}\mid n\in \mathbb {N}\right \} .$ Let $D:=\left \{ \frac {1}{n}\mid n\in \mathbb {N}\right \} .$ Then $D=D_{1}\cup D_{2}.$ Consider the function $f:D\to \mathbb {R}$ determined by

  $$ f(x):=\begin{cases} 1 & \text{if }x\in D_{1}\\ 0 & \text{if }x\in D_{2} \end{cases}. $$

  Show $0$ is an accumulation point of $D_{1}$ and of $D_{2}$ and show that $\lim _{x\to 0}f(x)$ does not exists.

1.5 Problems for Sect. 1.5

• 1. Let $f(x)=\frac {x}{|2x+3|}.$ Find both limits at infinity: $\lim _{x\to \infty }f(x)$ and $\lim _{x\to -\infty }f(x).$

• 2. If $N\in \mathbb {N},$ prove $\lim _{x\to \infty }x^{n}=\infty .$ What happens as $x\to -\infty ?$

• 3. If $f$ and $g$ are real valued and $f(x)\leq g(x)$ for all $x,$ then

  $$ \lim_{x\to a}f(x)=\infty\implies\lim_{x\to a}g(x)=\infty. $$

• 4. If $f$ and $g$ are real valued, then

  $$ \lim_{x\to a}f(x)=4\text{ and }\lim_{x\to a}g(x)=\infty\implies\lim_{x\to a}f(x)g(x)=\infty. $$

  Give a proof based on the definitions. In particular, do not use the product rule for limits or use $4\cdot \infty =\infty .$

• 5. If $f\left (x\right )>0$ for all $x$ and $f\left (x\right )\to 0$ as $x\to a,$ then $\frac {1}{f\left (x\right )}\to \infty$ as $x\to a.$ Give a proof based on the definitions.

1.6 Problems for Sect. 1.6

• 1. If $(x_{n})$ is bounded and $(y_{n})$ is null, then $(x_{n}y_{n})$ is null.

• 2. If $(y_{n})$ is null and $|x_{n}|\leq |y_{n}|,$ then $(x_{n})$ is null.

• 3. If $x_{n}\leq y_{n}$ and $x_{n}\to \infty ,$ then $y_{n}\to \infty .$

• 4. If $z_{n}\neq 0$ for all $N,$ then $(z_{n})$ is null iff $1/|z_{n}|\to \infty .$

• 5. Find two sequences $(x_{n})$ and $(y_{n})$ such that $x_{n}\to \infty ,$ $y_{n}\to 0$ and $x_{n}y_{n}\to 1.$

• 6. Find two sequences $(x_{n})$ and $(y_{n})$ such that $x_{n}\to \infty ,$ $y_{n}\to 0$ and $x_{n}y_{n}\to 29.$

• 7. Find two sequences $(x_{n})$ and $(y_{n})$ such that $x_{n}\to \infty ,$ $y_{n}\to \infty$ and $x_{n}-y_{n}\to 1.$

1.7 Problems for Sect. 1.7

• 1. Let $x\in \mathbb {R}.$ Suppose $1<x.$ Given any $M>0,$ prove there is an $N\in \mathbb {N},$ such that $n\geq N$ $\implies$ $M\leq x^{n}.$ That is prove $x^{n}\to \infty$ as $n\to \infty .$

• 2. Find the sum of the geometric series

  $$ 0.999\cdots=\sum_{k=1}^{\infty}\frac{9}{10^{k}}. $$

1.8 Problems for Sect. 1.8

• 1. Prove Steinhaus’ Three Distance Conjecture for $\phi =\tfrac {3}{7}.$

• 2. Investigate Steinhaus’ Three Distance Conjecture on the unit circle (Fig. 1.10).

  Fig. 1.10

  

  Circle version of Fig. 1.9

• 3. If $\phi$ and $\psi$ are irrationals and $M,N$ are positive integers, let

  $$\begin{aligned} 0= & \left\{ j_{0}\phi\right\} <\left\{ j_{1}\phi\right\} <\cdots<\left\{ j_{M}\phi\right\} <\left\{ j_{M+1}\phi\right\} =1\\ 0= & \left\{ k_{0}\psi\right\} <\left\{ k_{1}\psi\right\} <\cdots<\left\{ k_{N}\psi\right\} <\left\{ k_{N+1}\psi\right\} =1 \end{aligned}$$

  as in (1.10). Here, $j_{M+1}:=1/\phi$ and $k_{N+1}:=1/\psi$ are notational devises introduced to simplify the notation below. One version of Steinhaus problem then is to consider the lengths of the sides of the rectangles whose vertices are at the points of intersections of the lines in Fig. 1.11. In this case the number of distances is at most six, by the theorem in Sect. 1.8. A linear ordering on the plane is introduced in the problems for Sect. E. This linear ordering imposes a linear ordering on the points $\left (\left \{ j_{m}\phi \right \} ,\left \{ k_{n}\psi \right \} \right )_{m,n}.$ This introduces some additional distances into the problem. Find an upper bound on the number of length of “intervals” corresponding to this ordering.

  Fig. 1.11

  

  A two dimensional Steinhaus problem. In the figure $M=5,$ $N=4,$ $\phi =e,$ and $\psi =5\pi$

• 4. A different version of the Steinhaus problem would be to consider the “intervals” obtained by using the linear order from the problems to Sect. E on the points

  $$ \left(\left\{ k_{m}\phi\right\} ,\left\{ j_{m}\psi\right\} \right)_{m=0}^{M+1}. $$

  In particular, $N=M,$ we are only considering $M$ of the $M^{2}$ points in the open square, and of the points on the boundary only $\left (\left \{ k_{0}\phi \right \} ,\left \{ j_{0}\psi \right \} \right )=(0,0)$ and $\left (\left \{ k_{M+1}\phi \right \} ,\left \{ j_{M+1}\psi \right \} \right )=(1,1)$ are included in the list. [The author has not attempted to solve this problem, it may be easy or it may be very difficult.]

Solutions and Hints for the Exercises

Exercise 1.1.7.. If $1/x$ is rational, then $1/x=p/q,$ for integers $p,q\neq 0.$ Hence $x=q/p$ contradicting that $x$ is not rational. The other cases are similar.

Exercise 1.2.2. Let $\varepsilon>0$ be given. If $\varepsilon \leq m,$ then $D\cap B'_{\varepsilon }(c)\neq \emptyset$ by assumption. On the other hand, if $m<\varepsilon ,$ then

$$ D\cap B'_{\varepsilon}(c)\supseteq D\cap B'_{m}(c) $$

and the right hand side is nonempty by assumption.

Exercise 1.3.9.. If $\left |x-1/2\right |<1/4,$ then $-1/4<x-1/2<1/4,$ and therefore $1/4<x<3/4.$ But $1/4<x,$ implies $0<2/x<8.$ Hence,

$$ \left|x-\frac{1}{2}\right|<\frac{1}{4}\implies\frac{2}{x}\left|x-\frac{1}{2}\right|<8\left|x-\frac{1}{2}\right|. $$

Consequently, if $|x-1/2|<1/4$ and $|x-1/2|<\varepsilon /8,$ then

$$ \frac{2}{x}\left|x-\frac{1}{2}\right|<8\left|x-\frac{1}{2}\right|<8\frac{\varepsilon}{8}=\varepsilon. $$

Thus, $\delta :=\min \{1/4,\varepsilon /8\}$ works.

Exercise 1.3.11.. (i). It is easy to justify:

$$\begin{aligned} g\left(\{x\mid0<|x-0|<\delta\}\right) & \supseteq\{g(x)\mid0<x<\delta\}\\ & =\{\sigma(1/x)\mid0<x<\delta\}\\ & =\{\sigma(t)\mid1/\delta<t\}\\ & \supseteq\{-1,1\}. \end{aligned}$$

(ii). For any real number $L,$ either $|L-1|\geq 1$ or $|L-(-1)|\geq 1.$

Exercise 1.3.15. This follows from

$$ |f(x)|=|L-\left(L+f(x)\right)|\geq|L|-|L-f(x)|. $$

An alternative argument is to show $f(x)\to L$ implies $|f(x)|\to |L|$ and then use Local Positivity on $g(x)=|f(x)|.$

Exercise 1.4.7.. Use Example 1.3.5, the Product Rule, and induction on $n.$

Exercise 1.4.8. One way is to use the Sum Rule and induction on the degree.

Exercise 1.4.10. This is similar to the last part of the proof of the Product Rule. We need to control the size of the factor $|1/g(x)|$, hence we need a lower bound on $|g(x)|.$ This is provided by Local Positivity in the form of Exercise 1.3.15.

Exercise 1.4.15. Comparing to the Composition Rule with $f(x)=1+x$ and $g(x)=\sqrt {x}.$ As $x\to 3$ we have $f(x)\to 1+3=4,$ and as $x\to 4$ we have $g(x)\to 2$, by assumption. To use the Composition Rule we need a $\gamma>0,$ such that $0<|x-3|<\gamma$ implies $f(x)=1+x\neq 4.$ Since $x\neq 3$ implies $1+x\neq 4$ any choice, e.g., $\gamma =1$ works.

Exercise 1.4.18. (i) $D_{1}\cap B_{r}'(a)\subseteq D\cap B_{r}'(a).$

(ii) If $D_{1}\cap B_{r}'(a)=D_{2}\cap B_{r}'(a)=\emptyset ,$ then $(D_{1}\cup D_{2})\cap B_{r}'(a)=\emptyset .$

Exercise 1.5.1. This is a special case of the Corollary in Sect. 1.4..

Exercise 1.5.4. In one case set $N=1/\varepsilon ^{1/n}$ in the other set $N=-1/\varepsilon ^{1/n}.$

Exercise 1.6.5. This is a special case of Theorem 1.3.14.

Exercise 1.6.8. Let $\varepsilon :=3/2.$ Given $N,$ there is an even integer $n>N.$ Since $n$ is even, $1+(-1)^{n}=2>3/2=\varepsilon .$

Exercise 1.7.2. (1) $1<1/x.$ (2) $\left (\frac {1}{x}\right )^{n}=(1+y)^{n}.$ (3) Rearrange (2). (4) Since $N>\varepsilon /y$ this follows from (3).