Real Analysis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 8 July 2014

Lecture 6

Graphing and limits

Graph $f\left(x\right)=\{\begin{array}{cc}\text{sin}\left(\frac{1}{x}\right),& \text{if}\hspace{0.17em}x\ne 0,\\ 1,& \text{if}\hspace{0.17em}x=0\text{.}\end{array}$ $$\begin{array}{c} x y 1 -1 -1π -12π 12π 1π 2π \end{array}$$ $\underset{x\to a}{\text{lim}}f\left(x\right)=\ell$ if $f\left(x\right)$ gets closer and closer to $\ell$ as $x$ gets closer and closer to $a\text{.}$

$\underset{x\to 0}{\text{lim}}\text{sin}\left(\frac{1}{x}\right)$ does not exist because $\text{sin}\left(\frac{1}{x}\right)$ oscillates between $0$ and $1$ as $x$ gets closer and closer to $0\text{.}$

The function $f\left(x\right)$ is continuous at $x=a$ if $f\left(x\right)$ is such that $$\underset{x\to a}{\text{lim}}f\left(x\right)=f\left(a\right)\text{.}$$ The function $f\left(x\right)=\{\begin{array}{cc}\text{sin}\left(\frac{1}{x}\right),& \text{if}\hspace{0.17em}x\ne 0,\\ 1,& \text{if}\hspace{0.17em}x=0\end{array}$ is not continuous at $x=a$ because $\underset{x\to 0}{\text{lim}}f\left(x\right)=\underset{x\to 0}{\text{lim}}\text{sin}\left(\frac{1}{x}\right)$ does not exist.

Graph $y=\text{tan}\hspace{0.17em}x=\frac{\text{sin}\hspace{0.17em}x}{\text{cos}\hspace{0.17em}x}\text{.}$ $$\begin{array}{c} - 2 π - 3 π 2 -π - π 2 π 2 π 3 π 2 2 π x -1 1 y \end{array}$$ Notes:

(a)	If $\text{sin}\hspace{0.17em}x=0$ then $y=0\text{.}$
(b)	If $\text{cos}\hspace{0.17em}x=0$ then $y$ is very large.

$\underset{x\to \frac{\pi }{2}}{\text{lim}}\text{tan}\hspace{0.17em}x$ does not exist because $\text{tan}\hspace{0.17em}x$ gets larger and larger as $x$ gets closer and closer to $\frac{\pi }{2}\text{.}$

Graph $y=\lfloor x\rfloor \text{.}$

$\lfloor x\rfloor =n,$ if $n\in \mathbb{Z}$ and $n\le x<n+1,$ i.e. $\lfloor x\rfloor$ is the largest integer $\le x\text{.}$ $$\begin{array}{c} - 3 - 2 - 1 1 2 3 4 x - 2 - 1 1 2 y \end{array}$$ $\underset{x\to 1}{\text{lim}}\lfloor x\rfloor$ does not exist because $$\underset{x\to 1^{-}}{\text{lim}}\lfloor x\rfloor =0\text{and}\underset{x\to 1^{+}}{\text{lim}}\lfloor x\rfloor =1,$$ where $\underset{x\to 1^{-}}{\text{lim}}\lfloor x\rfloor$ is the limit of $\lfloor x\rfloor$ as $x$ gets closer and closer to $1$ from the negative side of $1,$ and $\underset{x\to 1^{+}}{\text{lim}}\lfloor x\rfloor$ is the limit of $\lfloor x\rfloor$ as $x$ gets closer and closer to $1$ from the positive side of $1\text{.}$

The function $f\left(x\right)=\lfloor x\rfloor ,$ for $x\in ℝ,$ is continuous for $x\notin \mathbb{Z}\text{.}$

In English: A function $f\left(x\right)$ is continuous at $x=a$ if $f\left(x\right)$ doesn't jump at $x=a\text{.}$

In math: A function $f\left(x\right)$ is continuous at $x=a$ if $\underset{x\to a}{\text{lim}}f\left(x\right)=f\left(a\right)\text{.}$

Let $z=\frac{1}{2}{e}^{i\pi /4}\text{.}$ Graph the sequence $a_n=z^n\text{.}$

A sequence in $ℂ$ is a function $$\begin{array}{ccc}{\mathbb{Z}}_{>0}& ⟶& ℂ\\ n& ⟼& a_n\end{array}$$ The first $7$ terms of this sequence are $$\begin{array}{c}{a}_1=\frac{1}{2}{e}^{i\pi /4},a_2=\frac{1}{4}{e}^{i2\pi /4},a_3=\frac{1}{8}{e}^{i3\pi /4},a_4=\frac{1}{16}{e}^{i4\pi /4},\\ a_5=\frac{1}{32}{e}^{i5\pi /4},a_6=\frac{1}{64}{e}^{i6\pi /4},a_7=\frac{1}{128}{e}^{i7\pi /4}\text{.}\end{array}$$ Graph $x+iy$ as $$\begin{array}{c} - 3 - 2 i 1 + 1 i x + iy 1 + 0 i x not i axis 0 + 1 i 0 + 2 i y i axis \end{array}$$ So $$\begin{array}{c}{a}_1=\frac{1}{2}{e}^{i\pi /4}=\frac{1}{2}(\text{cos}\hspace{0.17em}\frac{\pi }{4}+i\text{sin}\hspace{0.17em}\frac{\pi }{4})=\frac{1}{2}(\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}),\\ a_2-\frac{1}{4}{e}^{i2\pi /4}-\frac{1}{4}{e}^{i\pi /2}=\frac{1}{4}(\text{cos}\hspace{0.17em}\frac{\pi }{2}+i\text{sin}\hspace{0.17em}\frac{\pi }{2})=\frac{1}{4}(0+i)\text{.}\end{array}$$ $$\begin{array}{c} a1 a2 a3 a4 a5 eiπ4 ei2π4 ei3π4 ei4π4 ei5π4 ei6π4 ei7π4 - 1 1 - i i \end{array}$$ $\underset{n\to \infty }{\text{lim}}{a}_n=\ell$ means $a_n$ gets closer and closer to $\ell$ as $n$ gets larger and larger.

In our example the $a_n$ are getting closer and closer to $0+0i,$ in a spiral.

Definition Let $z=x+iy$ be in $ℂ\text{.}$ The absolute value of $z$ is $$\left|z\right|=\sqrt{x^2+y^2}\text{.}$$ In English: $\left|z\right|$ is the distance from $z$ to $0+0i\text{.}$

The complex numbers is $$ℂ=\{x+iy\hspace{0.17em}|\hspace{0.17em}x,y\in ℝ\}\text{with}{i}^2=-1\text{.}$$

Notes and References

These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100312Lect6.pdf and was given on 11 March 2010.

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