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 9
Limit examples
Prove that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
So
|
Evaluate
Find
since
and
exist (because
and
are continuous at – the graph doesn't jump at
Find
NO. We can only do this if
exists and exists.
A more sensible approach is to use some algebra,
Evaluate
Evaluate
Let Then
and
as So
Evaluate
Let and be sequences in
Assume that exists
and exists. If
then
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Proof by contradiction.
Let
and
Assume Let
Let be such that
if and
then
Let be such that
if and
then
Let be such that
and
Then
This is a contradiction to
So
|
Assume that
and
exists. Then
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Let
To show:
To show: If then there exists such that
if then
Assume
To show: There exists such that
if then
Let be such that
if
then
Let be such that
if then
So
if then
|
Notes and References
These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100319suggLect9.pdf and was given on 19 March 2010.
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