Problem with multivariable inequality

Clash Royale CLAN TAG#URR8PPP

up vote
1
down vote

favorite

Basically I'm trying to prove a limit with the delta epsilon definition and I'm reading about how to do it. In one example, the book does the following:

$lim_(x,y)to (1,-1) fracy+1(3(x-1)^2+2(y+1)^2)^1/3 leq lim_(x,y)to (1,-1) fraclVert[x-1,y+1]rVert(2lVert[x-1,y+1]rVert ^2+2lVert[x-1,y+1]rVert^2)^1/3$

I don't understand why this is true. I know for a fact that $(y+1) leq ||[x-1,y+1]||$ but I don't understand what's going on in the denominator. As far as I know the denominator of the first function should be greater than or equal to the one on the second function for this inequality to hold true, but I don't understand why it works here.

Edit: $lVert [x-1,y+1] rVert^2$ is equal to $sqrt (x-1)^2+(y+1)^2 $

analysis inequality

share|cite|improve this question

edited Jul 14 at 21:59

asked Jul 14 at 20:52

Direwolfox

62

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  What is $||(x-1),(y+1)||$?
  – Mostafa Ayaz
  Jul 14 at 20:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The right hand side is the sum of two identical terms?
  – Bernard
  Jul 14 at 21:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @MostafaAyaz Oh sorry it is the norm of the vector. I already edited my notation to the one in MathJax.
  – Direwolfox
  Jul 14 at 21:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Bernard Yes it is, I guess it is written that way to make it more understandable, as in one of those is related to 3(x-1)^2 and the other one to 2(y+1)^2. In the next step he adds them up and ends up with 4||(x-1),(y+1)||^2
  – Direwolfox
  Jul 14 at 21:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Direwolfox do you mean a vector with entries $[x-1,y+1]$?
  – mathreadler
  Jul 14 at 21:14
  
  

  
  
  
  

 | 
show 3 more comments

up vote
1
down vote

favorite

Basically I'm trying to prove a limit with the delta epsilon definition and I'm reading about how to do it. In one example, the book does the following:

$lim_(x,y)to (1,-1) fracy+1(3(x-1)^2+2(y+1)^2)^1/3 leq lim_(x,y)to (1,-1) fraclVert[x-1,y+1]rVert(2lVert[x-1,y+1]rVert ^2+2lVert[x-1,y+1]rVert^2)^1/3$

I don't understand why this is true. I know for a fact that $(y+1) leq ||[x-1,y+1]||$ but I don't understand what's going on in the denominator. As far as I know the denominator of the first function should be greater than or equal to the one on the second function for this inequality to hold true, but I don't understand why it works here.

Edit: $lVert [x-1,y+1] rVert^2$ is equal to $sqrt (x-1)^2+(y+1)^2 $

analysis inequality

share|cite|improve this question

edited Jul 14 at 21:59

asked Jul 14 at 20:52

Direwolfox

62

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  What is $||(x-1),(y+1)||$?
  – Mostafa Ayaz
  Jul 14 at 20:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The right hand side is the sum of two identical terms?
  – Bernard
  Jul 14 at 21:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @MostafaAyaz Oh sorry it is the norm of the vector. I already edited my notation to the one in MathJax.
  – Direwolfox
  Jul 14 at 21:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Bernard Yes it is, I guess it is written that way to make it more understandable, as in one of those is related to 3(x-1)^2 and the other one to 2(y+1)^2. In the next step he adds them up and ends up with 4||(x-1),(y+1)||^2
  – Direwolfox
  Jul 14 at 21:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Direwolfox do you mean a vector with entries $[x-1,y+1]$?
  – mathreadler
  Jul 14 at 21:14
  
  

  
  
  
  

 | 
show 3 more comments

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Basically I'm trying to prove a limit with the delta epsilon definition and I'm reading about how to do it. In one example, the book does the following:

$lim_(x,y)to (1,-1) fracy+1(3(x-1)^2+2(y+1)^2)^1/3 leq lim_(x,y)to (1,-1) fraclVert[x-1,y+1]rVert(2lVert[x-1,y+1]rVert ^2+2lVert[x-1,y+1]rVert^2)^1/3$

I don't understand why this is true. I know for a fact that $(y+1) leq ||[x-1,y+1]||$ but I don't understand what's going on in the denominator. As far as I know the denominator of the first function should be greater than or equal to the one on the second function for this inequality to hold true, but I don't understand why it works here.

Edit: $lVert [x-1,y+1] rVert^2$ is equal to $sqrt (x-1)^2+(y+1)^2 $

analysis inequality

share|cite|improve this question

edited Jul 14 at 21:59

asked Jul 14 at 20:52

Direwolfox

62

Basically I'm trying to prove a limit with the delta epsilon definition and I'm reading about how to do it. In one example, the book does the following:

$lim_(x,y)to (1,-1) fracy+1(3(x-1)^2+2(y+1)^2)^1/3 leq lim_(x,y)to (1,-1) fraclVert[x-1,y+1]rVert(2lVert[x-1,y+1]rVert ^2+2lVert[x-1,y+1]rVert^2)^1/3$

I don't understand why this is true. I know for a fact that $(y+1) leq ||[x-1,y+1]||$ but I don't understand what's going on in the denominator. As far as I know the denominator of the first function should be greater than or equal to the one on the second function for this inequality to hold true, but I don't understand why it works here.

Edit: $lVert [x-1,y+1] rVert^2$ is equal to $sqrt (x-1)^2+(y+1)^2 $

analysis inequality

share|cite|improve this question

edited Jul 14 at 21:59

asked Jul 14 at 20:52

Direwolfox

62

share|cite|improve this question

share|cite|improve this question

edited Jul 14 at 21:59

edited Jul 14 at 21:59

edited Jul 14 at 21:59

asked Jul 14 at 20:52

Direwolfox

62

asked Jul 14 at 20:52

Direwolfox

62

asked Jul 14 at 20:52

Direwolfox

62

62

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  What is $||(x-1),(y+1)||$?
  – Mostafa Ayaz
  Jul 14 at 20:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The right hand side is the sum of two identical terms?
  – Bernard
  Jul 14 at 21:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @MostafaAyaz Oh sorry it is the norm of the vector. I already edited my notation to the one in MathJax.
  – Direwolfox
  Jul 14 at 21:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Bernard Yes it is, I guess it is written that way to make it more understandable, as in one of those is related to 3(x-1)^2 and the other one to 2(y+1)^2. In the next step he adds them up and ends up with 4||(x-1),(y+1)||^2
  – Direwolfox
  Jul 14 at 21:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Direwolfox do you mean a vector with entries $[x-1,y+1]$?
  – mathreadler
  Jul 14 at 21:14
  
  

  
  
  
  

 | 
show 3 more comments

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  What is $||(x-1),(y+1)||$?
  – Mostafa Ayaz
  Jul 14 at 20:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The right hand side is the sum of two identical terms?
  – Bernard
  Jul 14 at 21:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @MostafaAyaz Oh sorry it is the norm of the vector. I already edited my notation to the one in MathJax.
  – Direwolfox
  Jul 14 at 21:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Bernard Yes it is, I guess it is written that way to make it more understandable, as in one of those is related to 3(x-1)^2 and the other one to 2(y+1)^2. In the next step he adds them up and ends up with 4||(x-1),(y+1)||^2
  – Direwolfox
  Jul 14 at 21:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Direwolfox do you mean a vector with entries $[x-1,y+1]$?
  – mathreadler
  Jul 14 at 21:14
  
  

  
  
  
  

1

1

What is $||(x-1),(y+1)||$?
– Mostafa Ayaz
Jul 14 at 20:56

What is $||(x-1),(y+1)||$?
– Mostafa Ayaz
Jul 14 at 20:56

The right hand side is the sum of two identical terms?
– Bernard
Jul 14 at 21:04

The right hand side is the sum of two identical terms?
– Bernard
Jul 14 at 21:04

@MostafaAyaz Oh sorry it is the norm of the vector. I already edited my notation to the one in MathJax.
– Direwolfox
Jul 14 at 21:12

@MostafaAyaz Oh sorry it is the norm of the vector. I already edited my notation to the one in MathJax.
– Direwolfox
Jul 14 at 21:12

@Bernard Yes it is, I guess it is written that way to make it more understandable, as in one of those is related to 3(x-1)^2 and the other one to 2(y+1)^2. In the next step he adds them up and ends up with 4||(x-1),(y+1)||^2
– Direwolfox
Jul 14 at 21:14

@Bernard Yes it is, I guess it is written that way to make it more understandable, as in one of those is related to 3(x-1)^2 and the other one to 2(y+1)^2. In the next step he adds them up and ends up with 4||(x-1),(y+1)||^2
– Direwolfox
Jul 14 at 21:14

@Direwolfox do you mean a vector with entries $[x-1,y+1]$?
– mathreadler
Jul 14 at 21:14

@Direwolfox do you mean a vector with entries $[x-1,y+1]$?
– mathreadler
Jul 14 at 21:14

 | 
show 3 more comments

active

oldest

votes

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2851955%2fproblem-with-multivariable-inequality%23new-answer', 'question_page');

);

Post as a guest

Name Email

active

oldest

votes

active

oldest

votes

active

oldest

votes

active

oldest

votes

 

draft saved

draft discarded

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2851955%2fproblem-with-multivariable-inequality%23new-answer', 'question_page');

);

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Name Email

Name

Email

Name Email

Name

Email