How to prove $lim_{(x,y) to (0,0)} frac{xy}{x^4+y^4}$ does not exist












1












$begingroup$


$$lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$$



Anyone can teach me how to solve this question? I have tried so many times but still unable to solve it.










share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    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.
    $endgroup$
    – Gerry Myerson
    Dec 2 '18 at 3:39










  • $begingroup$
    Try to choose a suitable path depending on some variable constant.
    $endgroup$
    – Anik Bhowmick
    Dec 2 '18 at 4:00










  • $begingroup$
    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
    $endgroup$
    – Rhys Hughes
    Dec 2 '18 at 4:21


















1












$begingroup$


$$lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$$



Anyone can teach me how to solve this question? I have tried so many times but still unable to solve it.










share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    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.
    $endgroup$
    – Gerry Myerson
    Dec 2 '18 at 3:39










  • $begingroup$
    Try to choose a suitable path depending on some variable constant.
    $endgroup$
    – Anik Bhowmick
    Dec 2 '18 at 4:00










  • $begingroup$
    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
    $endgroup$
    – Rhys Hughes
    Dec 2 '18 at 4:21
















1












1








1





$begingroup$


$$lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$$



Anyone can teach me how to solve this question? I have tried so many times but still unable to solve it.










share|cite|improve this question











$endgroup$




$$lim_{(x,y)to(0,0)} frac{xy}{x^4+y^4}$$



Anyone can teach me how to solve this question? I have tried so many times but still unable to solve it.







calculus limits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 2 '18 at 4:20









Tianlalu

3,08121038




3,08121038










asked Dec 2 '18 at 3:35









AlvinAlvin

61




61








  • 3




    $begingroup$
    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.
    $endgroup$
    – Gerry Myerson
    Dec 2 '18 at 3:39










  • $begingroup$
    Try to choose a suitable path depending on some variable constant.
    $endgroup$
    – Anik Bhowmick
    Dec 2 '18 at 4:00










  • $begingroup$
    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
    $endgroup$
    – Rhys Hughes
    Dec 2 '18 at 4:21
















  • 3




    $begingroup$
    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.
    $endgroup$
    – Gerry Myerson
    Dec 2 '18 at 3:39










  • $begingroup$
    Try to choose a suitable path depending on some variable constant.
    $endgroup$
    – Anik Bhowmick
    Dec 2 '18 at 4:00










  • $begingroup$
    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
    $endgroup$
    – Rhys Hughes
    Dec 2 '18 at 4:21










3




3




$begingroup$
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.
$endgroup$
– Gerry Myerson
Dec 2 '18 at 3:39




$begingroup$
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.
$endgroup$
– Gerry Myerson
Dec 2 '18 at 3:39












$begingroup$
Try to choose a suitable path depending on some variable constant.
$endgroup$
– Anik Bhowmick
Dec 2 '18 at 4:00




$begingroup$
Try to choose a suitable path depending on some variable constant.
$endgroup$
– Anik Bhowmick
Dec 2 '18 at 4:00












$begingroup$
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
$endgroup$
– Rhys Hughes
Dec 2 '18 at 4:21






$begingroup$
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
$endgroup$
– Rhys Hughes
Dec 2 '18 at 4:21












1 Answer
1






active

oldest

votes


















1












$begingroup$

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 $(0,0)$.






share|cite|improve this answer











$endgroup$













    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
    });


    }
    });














    draft saved

    draft discarded


















    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









    1












    $begingroup$

    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 $(0,0)$.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      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 $(0,0)$.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        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 $(0,0)$.






        share|cite|improve this answer











        $endgroup$



        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 $(0,0)$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 2 '18 at 4:45

























        answered Dec 2 '18 at 4:35









        UserSUserS

        1,5391112




        1,5391112






























            draft saved

            draft discarded




















































            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.




            draft saved


            draft discarded














            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





















































            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







            Popular posts from this blog

            Ellipse (mathématiques)

            Quarter-circle Tiles

            Mont Emei