Proving a Limit Does Not Exist
$begingroup$
I'm not even sure how to approach this. I try factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$ but I don't think that gets me anywhere with the denominator.
calculus limits multivariable-calculus
edited 29 mins ago
Thomas Shelby
3,1771524
asked 42 mins ago
krauser126krauser126
394
$endgroup$
add a comment |
$begingroup$
I'm not even sure how to approach this. I try factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$ but I don't think that gets me anywhere with the denominator.
calculus limits multivariable-calculus
edited 29 mins ago
Thomas Shelby
3,1771524
asked 42 mins ago
krauser126krauser126
394
$endgroup$
add a comment |
$begingroup$
I'm not even sure how to approach this. I try factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$ but I don't think that gets me anywhere with the denominator.
calculus limits multivariable-calculus
edited 29 mins ago
Thomas Shelby
3,1771524
asked 42 mins ago
krauser126krauser126
394
$endgroup$
I'm not even sure how to approach this. I try factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$ but I don't think that gets me anywhere with the denominator.
calculus limits multivariable-calculus
calculus limits multivariable-calculus
edited 29 mins ago
Thomas Shelby
3,1771524
asked 42 mins ago
krauser126krauser126
394
edited 29 mins ago
Thomas Shelby
3,1771524
asked 42 mins ago
krauser126krauser126
394
edited 29 mins ago
Thomas Shelby
3,1771524
edited 29 mins ago
Thomas Shelby
3,1771524
edited 29 mins ago
Thomas Shelby
3,1771524
3,1771524
asked 42 mins ago
krauser126krauser126
394
asked 42 mins ago
krauser126krauser126
394
asked 42 mins ago
krauser126krauser126
394
394
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
answered 38 mins ago
Mark ViolaMark Viola
132k1275173
$endgroup$
add a comment |
$begingroup$
Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
answered 38 mins ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
71729
$endgroup$
add a comment |
$begingroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
&lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}\
&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
edited 1 min ago
Javi maxwell
878
answered 33 mins ago
Thomas ShelbyThomas Shelby
3,1771524
$endgroup$
add a comment |
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',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3111000%2fproving-a-limit-does-not-exist%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
answered 38 mins ago
Mark ViolaMark Viola
132k1275173
$endgroup$
add a comment |
$begingroup$
HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
answered 38 mins ago
Mark ViolaMark Viola
132k1275173
$endgroup$
add a comment |
$begingroup$
HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
answered 38 mins ago
Mark ViolaMark Viola
132k1275173
$endgroup$
HINT:
What happens if the limit is taken along $y=2x$? What happens when the limit is taken along $y=0$? Are these equal? If not, what can one conclude?
answered 38 mins ago
Mark ViolaMark Viola
132k1275173
answered 38 mins ago
Mark ViolaMark Viola
132k1275173
answered 38 mins ago
Mark ViolaMark Viola
132k1275173
answered 38 mins ago
Mark ViolaMark Viola
132k1275173
132k1275173
add a comment |
add a comment |
$begingroup$
Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
answered 38 mins ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
71729
$endgroup$
add a comment |
$begingroup$
Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
answered 38 mins ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
71729
$endgroup$
add a comment |
$begingroup$
Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
answered 38 mins ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
71729
$endgroup$
Putting $x=y$ your expression vanishes, and for$x=2y$, the limit will be $1/3$. Therefore the limit does not exist
answered 38 mins ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
71729
answered 38 mins ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
71729
answered 38 mins ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
71729
answered 38 mins ago
HAMIDINE SOUMAREHAMIDINE SOUMARE
71729
71729
add a comment |
add a comment |
$begingroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
&lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}\
&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
edited 1 min ago
Javi maxwell
878
answered 33 mins ago
Thomas ShelbyThomas Shelby
3,1771524
$endgroup$
add a comment |
$begingroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
&lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}\
&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
edited 1 min ago
Javi maxwell
878
answered 33 mins ago
Thomas ShelbyThomas Shelby
3,1771524
$endgroup$
add a comment |
$begingroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
&lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}\
&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
edited 1 min ago
Javi maxwell
878
answered 33 mins ago
Thomas ShelbyThomas Shelby
3,1771524
$endgroup$
Let's approach the limit along the line $y=mx.$
$begin{align}
&lim_{(x,y)to (0,0)}dfrac{x^3y-xy^3}{x^4+2y^4}\
&=lim_{xto 0}dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\
&=lim_{xto 0}dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\
&=lim_{xto 0}dfrac{m-m^3}{1+2m^4}\
&=dfrac{m-m^3}{1+2m^4}\
end{align}$
So what can you conclude about the limit ?
edited 1 min ago
Javi maxwell
878
answered 33 mins ago
Thomas ShelbyThomas Shelby
3,1771524
edited 1 min ago
Javi maxwell
878
edited 1 min ago
Javi maxwell
878
edited 1 min ago
Javi maxwell
878
878
answered 33 mins ago
Thomas ShelbyThomas Shelby
3,1771524
answered 33 mins ago
Thomas ShelbyThomas Shelby
3,1771524
answered 33 mins ago
Thomas ShelbyThomas Shelby
3,1771524
3,1771524
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3111000%2fproving-a-limit-does-not-exist%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
