Why Can L'Hospital's Rule Not be Applied to the Sum or Difference of Limits?











up vote
1
down vote

favorite












Consider $$lim_{xto infty}frac{f(x)}{g(x)} + lim_{xto infty}frac{h(x)}{i(x)}$$
I was told that I cannot apply L'Hospital's to each individual limit and then join the limits as
$$lim_{xto infty}frac{f(x)}{g(x)} +frac{h(x)}{i(x)}$$
Why is this incorrect?






calculus limits







share|cite|improve this question




















  • 1




    Limit operator is not a linear transformation, that is why.
    – Bertrand Wittgenstein's Ghost
    3 hours ago












  • I think it only goes wrong when you will end up with something of the form $+infty - infty$. So, if you avoid these cases you can apply it if I am correct.
    – Stan Tendijck
    2 hours ago















up vote
1
down vote

favorite












Consider $$lim_{xto infty}frac{f(x)}{g(x)} + lim_{xto infty}frac{h(x)}{i(x)}$$
I was told that I cannot apply L'Hospital's to each individual limit and then join the limits as
$$lim_{xto infty}frac{f(x)}{g(x)} +frac{h(x)}{i(x)}$$
Why is this incorrect?






calculus limits







share|cite|improve this question




















  • 1




    Limit operator is not a linear transformation, that is why.
    – Bertrand Wittgenstein's Ghost
    3 hours ago












  • I think it only goes wrong when you will end up with something of the form $+infty - infty$. So, if you avoid these cases you can apply it if I am correct.
    – Stan Tendijck
    2 hours ago













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Consider $$lim_{xto infty}frac{f(x)}{g(x)} + lim_{xto infty}frac{h(x)}{i(x)}$$
I was told that I cannot apply L'Hospital's to each individual limit and then join the limits as
$$lim_{xto infty}frac{f(x)}{g(x)} +frac{h(x)}{i(x)}$$
Why is this incorrect?






calculus limits







share|cite|improve this question















Consider $$lim_{xto infty}frac{f(x)}{g(x)} + lim_{xto infty}frac{h(x)}{i(x)}$$
I was told that I cannot apply L'Hospital's to each individual limit and then join the limits as
$$lim_{xto infty}frac{f(x)}{g(x)} +frac{h(x)}{i(x)}$$
Why is this incorrect?






calculus limits




calculus limits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 hours ago









Bernard

116k637108




116k637108










asked 3 hours ago









Danielle

1457




1457








  • 1




    Limit operator is not a linear transformation, that is why.
    – Bertrand Wittgenstein's Ghost
    3 hours ago












  • I think it only goes wrong when you will end up with something of the form $+infty - infty$. So, if you avoid these cases you can apply it if I am correct.
    – Stan Tendijck
    2 hours ago














  • 1




    Limit operator is not a linear transformation, that is why.
    – Bertrand Wittgenstein's Ghost
    3 hours ago












  • I think it only goes wrong when you will end up with something of the form $+infty - infty$. So, if you avoid these cases you can apply it if I am correct.
    – Stan Tendijck
    2 hours ago








1




1




Limit operator is not a linear transformation, that is why.
– Bertrand Wittgenstein's Ghost
3 hours ago






Limit operator is not a linear transformation, that is why.
– Bertrand Wittgenstein's Ghost
3 hours ago














I think it only goes wrong when you will end up with something of the form $+infty - infty$. So, if you avoid these cases you can apply it if I am correct.
– Stan Tendijck
2 hours ago




I think it only goes wrong when you will end up with something of the form $+infty - infty$. So, if you avoid these cases you can apply it if I am correct.
– Stan Tendijck
2 hours ago










2 Answers
2






active

oldest

votes

















up vote
8
down vote



accepted










Consider the following example:



$$
lim_{xrightarrowinfty}frac{x^2}{x}+lim_{xrightarrowinfty}frac{-x^2}{x} = infty - infty = text{Undefined}
$$
$$
lim_{xrightarrowinfty}frac{x^2}{x}+frac{-x^2}{x} = 0
$$






share|cite|improve this answer








New contributor




JDMan4444 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

























    up vote
    1
    down vote













    The following identity



    $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)}{g(x)} + lim_{xto infty}frac{h(x)}{i(x)}$$



    doesn't hold in general and to solve the LHS limit by l'Hopital, if necessary, we need to put it in the form



    $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)i(x)+h(x)g(x)}{g(x)i(x)} $$



    the reason is that the case you are referring to is not among the cases considered by l'Hopital theorem.






    share|cite|improve this answer





















      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: 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%2f3017962%2fwhy-can-lhospitals-rule-not-be-applied-to-the-sum-or-difference-of-limits%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      8
      down vote



      accepted










      Consider the following example:



      $$
      lim_{xrightarrowinfty}frac{x^2}{x}+lim_{xrightarrowinfty}frac{-x^2}{x} = infty - infty = text{Undefined}
      $$
      $$
      lim_{xrightarrowinfty}frac{x^2}{x}+frac{-x^2}{x} = 0
      $$






      share|cite|improve this answer








      New contributor




      JDMan4444 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






















        up vote
        8
        down vote



        accepted










        Consider the following example:



        $$
        lim_{xrightarrowinfty}frac{x^2}{x}+lim_{xrightarrowinfty}frac{-x^2}{x} = infty - infty = text{Undefined}
        $$
        $$
        lim_{xrightarrowinfty}frac{x^2}{x}+frac{-x^2}{x} = 0
        $$






        share|cite|improve this answer








        New contributor




        JDMan4444 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.




















          up vote
          8
          down vote



          accepted







          up vote
          8
          down vote



          accepted






          Consider the following example:



          $$
          lim_{xrightarrowinfty}frac{x^2}{x}+lim_{xrightarrowinfty}frac{-x^2}{x} = infty - infty = text{Undefined}
          $$
          $$
          lim_{xrightarrowinfty}frac{x^2}{x}+frac{-x^2}{x} = 0
          $$






          share|cite|improve this answer








          New contributor




          JDMan4444 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          Consider the following example:



          $$
          lim_{xrightarrowinfty}frac{x^2}{x}+lim_{xrightarrowinfty}frac{-x^2}{x} = infty - infty = text{Undefined}
          $$
          $$
          lim_{xrightarrowinfty}frac{x^2}{x}+frac{-x^2}{x} = 0
          $$







          share|cite|improve this answer








          New contributor




          JDMan4444 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          share|cite|improve this answer



          share|cite|improve this answer






          New contributor




          JDMan4444 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          answered 3 hours ago









          JDMan4444

          2142




          2142




          New contributor




          JDMan4444 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.





          New contributor





          JDMan4444 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.






          JDMan4444 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.






















              up vote
              1
              down vote













              The following identity



              $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)}{g(x)} + lim_{xto infty}frac{h(x)}{i(x)}$$



              doesn't hold in general and to solve the LHS limit by l'Hopital, if necessary, we need to put it in the form



              $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)i(x)+h(x)g(x)}{g(x)i(x)} $$



              the reason is that the case you are referring to is not among the cases considered by l'Hopital theorem.






              share|cite|improve this answer

























                up vote
                1
                down vote













                The following identity



                $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)}{g(x)} + lim_{xto infty}frac{h(x)}{i(x)}$$



                doesn't hold in general and to solve the LHS limit by l'Hopital, if necessary, we need to put it in the form



                $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)i(x)+h(x)g(x)}{g(x)i(x)} $$



                the reason is that the case you are referring to is not among the cases considered by l'Hopital theorem.






                share|cite|improve this answer























                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  The following identity



                  $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)}{g(x)} + lim_{xto infty}frac{h(x)}{i(x)}$$



                  doesn't hold in general and to solve the LHS limit by l'Hopital, if necessary, we need to put it in the form



                  $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)i(x)+h(x)g(x)}{g(x)i(x)} $$



                  the reason is that the case you are referring to is not among the cases considered by l'Hopital theorem.






                  share|cite|improve this answer












                  The following identity



                  $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)}{g(x)} + lim_{xto infty}frac{h(x)}{i(x)}$$



                  doesn't hold in general and to solve the LHS limit by l'Hopital, if necessary, we need to put it in the form



                  $$lim_{xto infty}left(frac{f(x)}{g(x)} +frac{h(x)}{i(x)}right)=lim_{xto infty}frac{f(x)i(x)+h(x)g(x)}{g(x)i(x)} $$



                  the reason is that the case you are referring to is not among the cases considered by l'Hopital theorem.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 3 hours ago









                  gimusi

                  88k74393




                  88k74393






























                       

                      draft saved


                      draft discarded



















































                       


                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3017962%2fwhy-can-lhospitals-rule-not-be-applied-to-the-sum-or-difference-of-limits%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

                      A CLEAN and SIMPLE way to add appendices to Table of Contents and bookmarks

                      Image
                      0 I found a lot of questions abount appendices and ToC. Many users want appendices to be grouped in an Appendix part, however some problems arise with ToC, hyperref, PDF viewer bookmarks, and so on. There are different solutions which require extra packages, command patching and other extra code, however none of them satisfies me. I almost found an easy way to accomplish a good result, where appendices are added to bookmarks in the right way and hyperref links point to the right page. However, the number of the "Appendix" part page is wrong (it's the number of appendix A). Is there any EASY way to fix that? This is a MWE: documentclass{book} usepackage[nottoc,notlot,notlof]{tocbibind} usepackage{hyperref} begin{document} frontmatter tableofcontents mainmatter part{First} chapter{
                      Read more

                      Calculate evaluation metrics using cross_val_predict sklearn

                      Image
                      1 In the sklearn.model_selection.cross_val_predict page it is stated: Generate cross-validated estimates for each input data point. It is not appropriate to pass these predictions into an evaluation metric. Can someone explain what does it mean? If this gives estimate of Y (y prediction) for every Y (true Y), why can't I calculate metrics such as RMSE or coefficient of determination using these results? python scikit-learn cross-validation share | improve this question edited Nov 28 '18 at 17:52 desertnaut 20.3k 7 43 79
                      Read more

                      Insert data from modal to MySQL (multiple modal on website)

                      Image
                      0 I have problem with inserting data to MySQL from modal. My modal: <a href="#" class="badge badge-pill badge-success">6 komentarzy</a> <a href="#" class="badge badge-pill badge-danger">brak komentarzy</a> <a data-toggle="modal" href="#add_desk_comm_{$desk_['desk_id']}" data-target="#add_desk_comm_{$desk_['desk_id']}" class="ediiit">(dodaj)</a> <div class="modal fade" id="add_desk_comm_{$desk_['desk_id']}" tabindex="-1" role="dialog" aria-labelledby="edit_printer" aria-hidden="true"> <div class="modal-dialog" role="document">
                      Read more