How to prove $lim_{(x,y) to (0,0)} frac{xy}{x^4+y^4}$ does not exist
$$lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$$
Anyone can teach me how to solve this question? I have tried so many times but still unable to solve it.
calculus limits
edited Dec 2 '18 at 4:20
Tianlalu
3,09621038
asked Dec 2 '18 at 3:35
Alvin
61
add a comment |
$$lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$$
Anyone can teach me how to solve this question? I have tried so many times but still unable to solve it.
calculus limits
edited Dec 2 '18 at 4:20
Tianlalu
3,09621038
asked Dec 2 '18 at 3:35
Alvin
61
3
Work out what happens as $xto0$ with $y$ held fixed at zero. Then work out what happens as you approach the origin along the curve $y=x^4$. Then write up what you have found, and post it as an answer.
– Gerry Myerson
Dec 2 '18 at 3:39
Try to choose a suitable path depending on some variable constant.
– Anik Bhowmick
Dec 2 '18 at 4:00
Well the fact that the expression has transposable variables (i.e. $f(x,y) equiv f(y,x)$) means that you just need to find $lim_{xto 0} f(x,x)$, or more precisely: $$lim_{xto 0}{2x^{-2}}$$ I think
– Rhys Hughes
Dec 2 '18 at 4:21
add a comment |
$$lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$$
Anyone can teach me how to solve this question? I have tried so many times but still unable to solve it.
calculus limits
edited Dec 2 '18 at 4:20
Tianlalu
3,09621038
asked Dec 2 '18 at 3:35
Alvin
61
$$lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$$
Anyone can teach me how to solve this question? I have tried so many times but still unable to solve it.
calculus limits
calculus limits
edited Dec 2 '18 at 4:20
Tianlalu
3,09621038
asked Dec 2 '18 at 3:35
Alvin
61
edited Dec 2 '18 at 4:20
Tianlalu
3,09621038
asked Dec 2 '18 at 3:35
Alvin
61
edited Dec 2 '18 at 4:20
Tianlalu
3,09621038
edited Dec 2 '18 at 4:20
Tianlalu
3,09621038
edited Dec 2 '18 at 4:20
Tianlalu
3,09621038
3,09621038
asked Dec 2 '18 at 3:35
Alvin
61
asked Dec 2 '18 at 3:35
Alvin
61
asked Dec 2 '18 at 3:35
Alvin
61
61
3
Work out what happens as $xto0$ with $y$ held fixed at zero. Then work out what happens as you approach the origin along the curve $y=x^4$. Then write up what you have found, and post it as an answer.
– Gerry Myerson
Dec 2 '18 at 3:39
Try to choose a suitable path depending on some variable constant.
– Anik Bhowmick
Dec 2 '18 at 4:00
Well the fact that the expression has transposable variables (i.e. $f(x,y) equiv f(y,x)$) means that you just need to find $lim_{xto 0} f(x,x)$, or more precisely: $$lim_{xto 0}{2x^{-2}}$$ I think
– Rhys Hughes
Dec 2 '18 at 4:21
add a comment |
3
Work out what happens as $xto0$ with $y$ held fixed at zero. Then work out what happens as you approach the origin along the curve $y=x^4$. Then write up what you have found, and post it as an answer.
– Gerry Myerson
Dec 2 '18 at 3:39
Try to choose a suitable path depending on some variable constant.
– Anik Bhowmick
Dec 2 '18 at 4:00
Well the fact that the expression has transposable variables (i.e. $f(x,y) equiv f(y,x)$) means that you just need to find $lim_{xto 0} f(x,x)$, or more precisely: $$lim_{xto 0}{2x^{-2}}$$ I think
– Rhys Hughes
Dec 2 '18 at 4:21
3
3
Work out what happens as $xto0$ with $y$ held fixed at zero. Then work out what happens as you approach the origin along the curve $y=x^4$. Then write up what you have found, and post it as an answer.
– Gerry Myerson
Dec 2 '18 at 3:39
Work out what happens as $xto0$ with $y$ held fixed at zero. Then work out what happens as you approach the origin along the curve $y=x^4$. Then write up what you have found, and post it as an answer.
– Gerry Myerson
Dec 2 '18 at 3:39
Try to choose a suitable path depending on some variable constant.
– Anik Bhowmick
Dec 2 '18 at 4:00
Try to choose a suitable path depending on some variable constant.
– Anik Bhowmick
Dec 2 '18 at 4:00
Well the fact that the expression has transposable variables (i.e. $f(x,y) equiv f(y,x)$) means that you just need to find $lim_{xto 0} f(x,x)$, or more precisely: $$lim_{xto 0}{2x^{-2}}$$ I think
– Rhys Hughes
Dec 2 '18 at 4:21
Well the fact that the expression has transposable variables (i.e. $f(x,y) equiv f(y,x)$) means that you just need to find $lim_{xto 0} f(x,x)$, or more precisely: $$lim_{xto 0}{2x^{-2}}$$ I think
– Rhys Hughes
Dec 2 '18 at 4:21
add a comment |
1 Answer
1
active
oldest
votes
Suppose the limit $lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$ exits then in some deleted nbd of $(0,0)$ the function $f:Bbb R^2-{(0,0)}rightarrow Bbb R,f(x,y)=frac{xy}{x^4+y^4}$ will be bounded . But notice that on the set $S:={(x,y)in Bbb R^2-{(0,0)}: x=y}$ the function $f$ is of the form $frac{x^2}{2x^4}=frac{1}{2x^2}$ i.e. in every deleted nbd of $(0,0)$ the function is unbounded.
$lim_{(x,y)rightarrow (0,0)} f(x,y)=limplies$ for $epsilon=1$ there is a $delta>0$ such that $0<x^2+y^2<deltaimplies |f(x,y)-l|<1implies l-1<f(x,y)<l+1 , forall (x,y)$ with $0<x^2+y^2<delta$ i.e. $f$ is bounded in a deleted nbd of $(0,0)$.
answered Dec 2 '18 at 4:35
UserS
1,5371112
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');
});
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%2f3022195%2fhow-to-prove-lim-x-y-to-0-0-fracxyx4y4-does-not-exist%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Suppose the limit $lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$ exits then in some deleted nbd of $(0,0)$ the function $f:Bbb R^2-{(0,0)}rightarrow Bbb R,f(x,y)=frac{xy}{x^4+y^4}$ will be bounded . But notice that on the set $S:={(x,y)in Bbb R^2-{(0,0)}: x=y}$ the function $f$ is of the form $frac{x^2}{2x^4}=frac{1}{2x^2}$ i.e. in every deleted nbd of $(0,0)$ the function is unbounded.
$lim_{(x,y)rightarrow (0,0)} f(x,y)=limplies$ for $epsilon=1$ there is a $delta>0$ such that $0<x^2+y^2<deltaimplies |f(x,y)-l|<1implies l-1<f(x,y)<l+1 , forall (x,y)$ with $0<x^2+y^2<delta$ i.e. $f$ is bounded in a deleted nbd of $(0,0)$.
answered Dec 2 '18 at 4:35
UserS
1,5371112
add a comment |
Suppose the limit $lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$ exits then in some deleted nbd of $(0,0)$ the function $f:Bbb R^2-{(0,0)}rightarrow Bbb R,f(x,y)=frac{xy}{x^4+y^4}$ will be bounded . But notice that on the set $S:={(x,y)in Bbb R^2-{(0,0)}: x=y}$ the function $f$ is of the form $frac{x^2}{2x^4}=frac{1}{2x^2}$ i.e. in every deleted nbd of $(0,0)$ the function is unbounded.
$lim_{(x,y)rightarrow (0,0)} f(x,y)=limplies$ for $epsilon=1$ there is a $delta>0$ such that $0<x^2+y^2<deltaimplies |f(x,y)-l|<1implies l-1<f(x,y)<l+1 , forall (x,y)$ with $0<x^2+y^2<delta$ i.e. $f$ is bounded in a deleted nbd of $(0,0)$.
answered Dec 2 '18 at 4:35
UserS
1,5371112
add a comment |
Suppose the limit $lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$ exits then in some deleted nbd of $(0,0)$ the function $f:Bbb R^2-{(0,0)}rightarrow Bbb R,f(x,y)=frac{xy}{x^4+y^4}$ will be bounded . But notice that on the set $S:={(x,y)in Bbb R^2-{(0,0)}: x=y}$ the function $f$ is of the form $frac{x^2}{2x^4}=frac{1}{2x^2}$ i.e. in every deleted nbd of $(0,0)$ the function is unbounded.
$lim_{(x,y)rightarrow (0,0)} f(x,y)=limplies$ for $epsilon=1$ there is a $delta>0$ such that $0<x^2+y^2<deltaimplies |f(x,y)-l|<1implies l-1<f(x,y)<l+1 , forall (x,y)$ with $0<x^2+y^2<delta$ i.e. $f$ is bounded in a deleted nbd of $(0,0)$.
answered Dec 2 '18 at 4:35
UserS
1,5371112
Suppose the limit $lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$ exits then in some deleted nbd of $(0,0)$ the function $f:Bbb R^2-{(0,0)}rightarrow Bbb R,f(x,y)=frac{xy}{x^4+y^4}$ will be bounded . But notice that on the set $S:={(x,y)in Bbb R^2-{(0,0)}: x=y}$ the function $f$ is of the form $frac{x^2}{2x^4}=frac{1}{2x^2}$ i.e. in every deleted nbd of $(0,0)$ the function is unbounded.
$lim_{(x,y)rightarrow (0,0)} f(x,y)=limplies$ for $epsilon=1$ there is a $delta>0$ such that $0<x^2+y^2<deltaimplies |f(x,y)-l|<1implies l-1<f(x,y)<l+1 , forall (x,y)$ with $0<x^2+y^2<delta$ i.e. $f$ is bounded in a deleted nbd of $(0,0)$.
answered Dec 2 '18 at 4:35
UserS
1,5371112
edited Dec 2 '18 at 4:45
answered Dec 2 '18 at 4:35
UserS
1,5371112
answered Dec 2 '18 at 4:35
UserS
1,5371112
answered Dec 2 '18 at 4:35
UserS
1,5371112
1,5371112
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3022195%2fhow-to-prove-lim-x-y-to-0-0-fracxyx4y4-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');
});
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');
});
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');
});
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
3
Work out what happens as $xto0$ with $y$ held fixed at zero. Then work out what happens as you approach the origin along the curve $y=x^4$. Then write up what you have found, and post it as an answer.
– Gerry Myerson
Dec 2 '18 at 3:39
Try to choose a suitable path depending on some variable constant.
– Anik Bhowmick
Dec 2 '18 at 4:00
Well the fact that the expression has transposable variables (i.e. $f(x,y) equiv f(y,x)$) means that you just need to find $lim_{xto 0} f(x,x)$, or more precisely: $$lim_{xto 0}{2x^{-2}}$$ I think
– Rhys Hughes
Dec 2 '18 at 4:21