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 7

Limits

$\underset{n\to \infty }{\text{lim}}{a}_n=\ell$ means $a_n$ gets closer and closer to $\ell$ as $n$ gets larger and larger.

$\underset{x\to a}{\text{lim}}f\left(x\right)=\ell$ means $f\left(x\right)$ gets closer and closer to $\ell$ as $x$ gets closer and closer to $a\text{.}$

Distance and absolute value

The absolute value on $ℝ$ is $$\begin{array}{ccc}ℝ& \to & {ℝ}_{\ge 0}\\ x& \mapsto & \left|x\right|\end{array}$$ given by $$\left|x\right|=\{\begin{array}{cc}x,& \text{if}\hspace{0.17em}x\in {ℝ}_{>0},\\ 0,& \text{if}\hspace{0.17em}x=0,\\ -x,& \text{if}\hspace{0.17em}x<0\text{.}\end{array}$$ The complex numbers is $$ℂ=\{x+iy\hspace{0.17em}|\hspace{0.17em}x,y\in ℝ\}\text{with}{i}^2=-1\text{.}$$ The absolute value on $ℂ$ is $$\begin{array}{ccc}ℂ& \to & {ℝ}_{\ge 0}\\ x& \mapsto & \left|x\right|\end{array}$$ given by $$|x+iy|=\left|\sqrt{x^2+y^2}\right|\text{.}$$ The vector space ${ℝ}^3$ is $${ℝ}^3=\left\{(x,y,z)\hspace{0.17em}\right|\hspace{0.17em}x,y,z\in ℝ\}\text{.}$$ The absolute value on ${ℝ}^3$ is $$\begin{array}{ccc}{ℝ}^3& \to & {ℝ}_{\ge 0}\\ v=(x,y,z)& \mapsto & \left|v\right|\end{array}$$ given by $$\left|v\right|=\left|\sqrt{x^2+y^2+z^2}\right|,\text{if}v=(x,y,z)\text{.}$$ So $$\begin{array}{c}\underset{n\to \infty }{\text{lim}}{a}_n=\ell \text{means}\underset{n\to \infty }{\text{lim}}|a_n-\ell |=0,\\ \underset{x\to a}{\text{lim}}f\left(x\right)=\ell \text{means}\underset{x\to a}{\text{lim}}|f\left(x\right)-\ell |=0\text{.}\end{array}$$

Official definitions in Maths

The sequence $\left(a_n\right)$ converges to $\ell$ if $\left(a_n\right)$ satisfies if $\epsilon \in {ℝ}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ such that
if $n\in {\mathbb{Z}}_{>0}$ and $n>N$ then $|a_n-\ell |<\epsilon \text{.}$ Write $\underset{n\to \infty }{\text{lim}}{a}_n=\ell$ if $\left(a_n\right)$ converges to $\ell \text{.}$

The function $f\left(x\right)$ converges to $\ell$ as $x\to a$ if $f\left(x\right)$ satisfies: if $\epsilon \in {ℝ}_{>0}$ then there exists $\delta \in {ℝ}_{>0}$ such that
if $|x-a|<\delta$ then $|f\left(x\right)-\ell |<\epsilon \text{.}$

Write $\underset{x\to a}{\text{lim}}f\left(x\right)=\ell$ if $f\left(x\right)$ converges to $\ell$ as $x\to a\text{.}$

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

MOST IMPORTANT PROPERTY of absolute value: $$|x+y|\le \left|x\right|+\left|y\right|\text{.}$$

Useful properties of limits

(a)

Assume that $\underset{n\to \infty }{\text{lim}}{a}_n$ and $\underset{n\to \infty }{\text{lim}}{b}_n$ exist. Then (a1) ${\displaystyle \underset{n\to \infty }{\text{lim}}(a_n+b_n)}={\displaystyle \underset{n\to \infty }{\text{lim}}{a}_n+\underset{n\to \infty }{\text{lim}}{b}_n,}$ (a2) ${\displaystyle \underset{n\to \infty }{\text{lim}}{a}_n{b}_n}={\displaystyle \left(\underset{n\to \infty }{\text{lim}}{a}_n\right)\left(\underset{n\to \infty }{\text{lim}}{b}_n\right),}$ (a3) ${\displaystyle \underset{n\to \infty }{\text{lim}}-a_n}={\displaystyle -\left(\underset{n\to \infty }{\text{lim}}{a}_n\right),}$ (a4) If $\left(a_n\right)$ satisfies: if $n\in {\mathbb{Z}}_{>0}$ then $a_n\ne 0,$ then $$\underset{n\to \infty }{\text{lim}}\frac{1}{a_n}=\frac{1}{\underset{n\to \infty }{\text{lim}}{a}_n}\text{.}$$

(b)

Assume that $\underset{x\to a}{\text{lim}}f\left(x\right)$ and $\underset{x\to a}{\text{lim}}g\left(x\right)$ exist. Then (b1) ${\displaystyle \underset{x\to a}{\text{lim}}(f\left(x\right)+g\left(x\right))}={\displaystyle \underset{x\to a}{\text{lim}}f\left(x\right)\underset{x\to a}{\text{lim}}g\left(x\right),}$ (b2) ${\displaystyle \underset{x\to a}{\text{lim}}\left(f\left(x\right)g\left(x\right)\right)}={\displaystyle \left(\underset{x\to a}{\text{lim}}f\left(x\right)\right)\left(\underset{x\to a}{\text{lim}}g\left(x\right)\right),}$ (b3) ${\displaystyle \underset{x\to a}{\text{lim}}(-f\left(x\right))}={\displaystyle -\underset{x\to a}{\text{lim}}f\left(x\right),}$ (b4) If $f\left(x\right)$ satisfies: $f\left(x\right)\ne 0$ for all $x$ close to $a$ then $$\underset{x\to a}{\text{lim}}\left(\frac{1}{f\left(x\right)}\right)=\frac{1}{\underset{x\to a}{\text{lim}}f\left(x\right)}\text{.}$$

(a)	Assume that $\left(a_n\right)$ and $\left(b_n\right)$ are sequences in $ℝ$ and $\underset{n\to \infty }{\text{lim}}{a}_n$ and $\underset{n\to \infty }{\text{lim}}{b}_n$ exist. If $a_n\le b_n$ then $\underset{n\to \infty }{\text{lim}}{a}_n\le $ \underset{n\to \infty }{\text{lim}}{b}_n$\text{.}$
(a')	Assume that $f\left(x\right)$ and $g\left(x\right)$ are real valued functions and $\underset{x\to a}{\text{lim}}f\left(x\right)$ and $\underset{x\to a}{\text{lim}}g\left(x\right)$ exist. If $f\left(x\right)\le g\left(x\right)$ then $\underset{x\to a}{\text{lim}}f\left(x\right)\le \underset{x\to a}{\text{lim}}g\left(x\right)\text{.}$
(b)	Assume that $\underset{x\to a}{\text{lim}}g\left(x\right)=\ell$ and $\underset{y\to \ell }{\text{lim}}f\left(y\right)$ exists.

Let $x\in ℂ\text{.}$ Then $\underset{y\to \ell }{\text{lim}}f\left(y\right)=\underset{x\to a}{\text{lim}}f\left(g\left(x\right)\right)\text{.}$

(a)	$\underset{n\to \infty }{\text{lim}}{x}^n=\{\begin{array}{cc}0,& \text{if}\hspace{0.17em}\left|x\right|<1,\\ \text{diverges,}& \text{if}\hspace{0.17em}\left|x\right|>1,\\ 1,& \text{if}\hspace{0.17em}x=1,\\ \text{diverges,}& \text{if}\hspace{0.17em}\left|x\right|=1\hspace{0.17em}\text{and}\hspace{0.17em}x\ne 1\text{.}\end{array}$
(b)	Let $n\in {\mathbb{Z}}_{>0}$ and $a\in ℂ\text{.}$ $\underset{x\to a}{\text{lim}}{x}^n=a^n$ (i.e. $f\left(x\right)=x^n$ is continuous).
(c)	Let $a\in ℂ\text{.}$ $\underset{x\to a}{\text{lim}}{e}^x=e^a$ (i.e. $f\left(x\right)=e^x$ is continuous).
(d)	$\underset{n\to \infty }{\text{lim}}1+x+x^2+\cdots +x^n=\{\begin{array}{cc}\frac{1}{1-x},& \text{if}\hspace{0.17em}\left|x\right|<1,\\ \text{diverges,}& \text{if}\hspace{0.17em}\left|x\right|\ge 1\text{.}\end{array}$
(e)	$\underset{n\to \infty }{\text{lim}}(1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!})$ exists.

Notes and References

These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100315Lect7.pdf and was given on 15 March 2010.

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