Does $lim_{x to - infty} left(frac{pi}{2} + arctan{x} right) cdot x = - infty$?












2












$begingroup$


Does $$lim_{x to - infty} left(frac{pi}{2} + arctan{x} right) cdot x = - infty$$?



My logic is that “something“ times "negative infinity" equals negative infinity. Am I right?






calculus limits infinity







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    The main flaw in your logic is that the "something" here happens to be $color{red}0$. (See points 2. and 3. of 5xum's answer, for example.)
    $endgroup$
    – Minus One-Twelfth
    Feb 27 at 10:11












  • $begingroup$
    @MinusOne-Twelfth Ah, I see it now. Thank you!
    $endgroup$
    – user644361
    Feb 27 at 10:13






  • 1




    $begingroup$
    You're welcome!
    $endgroup$
    – Minus One-Twelfth
    Feb 27 at 10:14
















2












$begingroup$


Does $$lim_{x to - infty} left(frac{pi}{2} + arctan{x} right) cdot x = - infty$$?



My logic is that “something“ times "negative infinity" equals negative infinity. Am I right?






calculus limits infinity







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    The main flaw in your logic is that the "something" here happens to be $color{red}0$. (See points 2. and 3. of 5xum's answer, for example.)
    $endgroup$
    – Minus One-Twelfth
    Feb 27 at 10:11












  • $begingroup$
    @MinusOne-Twelfth Ah, I see it now. Thank you!
    $endgroup$
    – user644361
    Feb 27 at 10:13






  • 1




    $begingroup$
    You're welcome!
    $endgroup$
    – Minus One-Twelfth
    Feb 27 at 10:14














2












2








2


0



$begingroup$


Does $$lim_{x to - infty} left(frac{pi}{2} + arctan{x} right) cdot x = - infty$$?



My logic is that “something“ times "negative infinity" equals negative infinity. Am I right?






calculus limits infinity







share|cite|improve this question











$endgroup$




Does $$lim_{x to - infty} left(frac{pi}{2} + arctan{x} right) cdot x = - infty$$?



My logic is that “something“ times "negative infinity" equals negative infinity. Am I right?






calculus limits infinity




calculus limits infinity






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Feb 27 at 11:26









Asaf Karagila♦

306k33437767




306k33437767










asked Feb 27 at 9:48









user644361user644361

835




835








  • 1




    $begingroup$
    The main flaw in your logic is that the "something" here happens to be $color{red}0$. (See points 2. and 3. of 5xum's answer, for example.)
    $endgroup$
    – Minus One-Twelfth
    Feb 27 at 10:11












  • $begingroup$
    @MinusOne-Twelfth Ah, I see it now. Thank you!
    $endgroup$
    – user644361
    Feb 27 at 10:13






  • 1




    $begingroup$
    You're welcome!
    $endgroup$
    – Minus One-Twelfth
    Feb 27 at 10:14














  • 1




    $begingroup$
    The main flaw in your logic is that the "something" here happens to be $color{red}0$. (See points 2. and 3. of 5xum's answer, for example.)
    $endgroup$
    – Minus One-Twelfth
    Feb 27 at 10:11












  • $begingroup$
    @MinusOne-Twelfth Ah, I see it now. Thank you!
    $endgroup$
    – user644361
    Feb 27 at 10:13






  • 1




    $begingroup$
    You're welcome!
    $endgroup$
    – Minus One-Twelfth
    Feb 27 at 10:14








1




1




$begingroup$
The main flaw in your logic is that the "something" here happens to be $color{red}0$. (See points 2. and 3. of 5xum's answer, for example.)
$endgroup$
– Minus One-Twelfth
Feb 27 at 10:11






$begingroup$
The main flaw in your logic is that the "something" here happens to be $color{red}0$. (See points 2. and 3. of 5xum's answer, for example.)
$endgroup$
– Minus One-Twelfth
Feb 27 at 10:11














$begingroup$
@MinusOne-Twelfth Ah, I see it now. Thank you!
$endgroup$
– user644361
Feb 27 at 10:13




$begingroup$
@MinusOne-Twelfth Ah, I see it now. Thank you!
$endgroup$
– user644361
Feb 27 at 10:13




1




1




$begingroup$
You're welcome!
$endgroup$
– Minus One-Twelfth
Feb 27 at 10:14




$begingroup$
You're welcome!
$endgroup$
– Minus One-Twelfth
Feb 27 at 10:14










3 Answers
3






active

oldest

votes


















6












$begingroup$

The sentence "something times negative infinity equals negative infinity" is wrong in more than one way:




  1. It is sloppy. Mathematics is rigorous for a reason. Mathematical statements must be well defined, and have a single meaning. In your case, what is "something"? Is an elephant times negative infinity equal to negative infinity?

  2. Even if you formalize the sentence, to something like "For all $alpha$, the limit $lim_{xto-infty}alphacdot x=-infty$, the statement remains false, since it is not true for any $alphaleq 0$.

  3. The sentence, applied to another limit, shows how wrong it is to use it to calculate limits. By your logic, $$lim_{xto-infty} frac{1}{x}cdot x=-infty,$$ since we have "something" times "negative infinity". Clearly, this is nonsence - the limit is obviously equal to $1$



The limit you must calculate is a limit of a product of two numbers, one tends to $0$, the other to $-infty$. Such a limit is often best approached using L'Hospital's rules.






share|cite|improve this answer











$endgroup$





















    2












    $begingroup$

    No. By L'Hopital's Rule $lim frac {frac {pi} 2+arctan, x} {1/x}=lim frac {1/(1+x^{2})} {-1/x^{2}}=-1$.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Yeah I have understood that. That's why I have deleted my comment.
      $endgroup$
      – Dbchatto67
      Feb 27 at 10:18





















    0












    $begingroup$

    Let $-1/x=h$



    $$lim_{hto0^+}dfrac{dfracpi2-arctandfrac1h}{-h}=-lim_{...}dfrac{arctan h}h=-1$$



    See Are $mathrm{arccot}(x)$ and $arctan(1/x)$ the same function?






    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%2f3128621%2fdoes-lim-x-to-infty-left-frac-pi2-arctanx-right-cdot-x%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









      6












      $begingroup$

      The sentence "something times negative infinity equals negative infinity" is wrong in more than one way:




      1. It is sloppy. Mathematics is rigorous for a reason. Mathematical statements must be well defined, and have a single meaning. In your case, what is "something"? Is an elephant times negative infinity equal to negative infinity?

      2. Even if you formalize the sentence, to something like "For all $alpha$, the limit $lim_{xto-infty}alphacdot x=-infty$, the statement remains false, since it is not true for any $alphaleq 0$.

      3. The sentence, applied to another limit, shows how wrong it is to use it to calculate limits. By your logic, $$lim_{xto-infty} frac{1}{x}cdot x=-infty,$$ since we have "something" times "negative infinity". Clearly, this is nonsence - the limit is obviously equal to $1$



      The limit you must calculate is a limit of a product of two numbers, one tends to $0$, the other to $-infty$. Such a limit is often best approached using L'Hospital's rules.






      share|cite|improve this answer











      $endgroup$


















        6












        $begingroup$

        The sentence "something times negative infinity equals negative infinity" is wrong in more than one way:




        1. It is sloppy. Mathematics is rigorous for a reason. Mathematical statements must be well defined, and have a single meaning. In your case, what is "something"? Is an elephant times negative infinity equal to negative infinity?

        2. Even if you formalize the sentence, to something like "For all $alpha$, the limit $lim_{xto-infty}alphacdot x=-infty$, the statement remains false, since it is not true for any $alphaleq 0$.

        3. The sentence, applied to another limit, shows how wrong it is to use it to calculate limits. By your logic, $$lim_{xto-infty} frac{1}{x}cdot x=-infty,$$ since we have "something" times "negative infinity". Clearly, this is nonsence - the limit is obviously equal to $1$



        The limit you must calculate is a limit of a product of two numbers, one tends to $0$, the other to $-infty$. Such a limit is often best approached using L'Hospital's rules.






        share|cite|improve this answer











        $endgroup$
















          6












          6








          6





          $begingroup$

          The sentence "something times negative infinity equals negative infinity" is wrong in more than one way:




          1. It is sloppy. Mathematics is rigorous for a reason. Mathematical statements must be well defined, and have a single meaning. In your case, what is "something"? Is an elephant times negative infinity equal to negative infinity?

          2. Even if you formalize the sentence, to something like "For all $alpha$, the limit $lim_{xto-infty}alphacdot x=-infty$, the statement remains false, since it is not true for any $alphaleq 0$.

          3. The sentence, applied to another limit, shows how wrong it is to use it to calculate limits. By your logic, $$lim_{xto-infty} frac{1}{x}cdot x=-infty,$$ since we have "something" times "negative infinity". Clearly, this is nonsence - the limit is obviously equal to $1$



          The limit you must calculate is a limit of a product of two numbers, one tends to $0$, the other to $-infty$. Such a limit is often best approached using L'Hospital's rules.






          share|cite|improve this answer











          $endgroup$



          The sentence "something times negative infinity equals negative infinity" is wrong in more than one way:




          1. It is sloppy. Mathematics is rigorous for a reason. Mathematical statements must be well defined, and have a single meaning. In your case, what is "something"? Is an elephant times negative infinity equal to negative infinity?

          2. Even if you formalize the sentence, to something like "For all $alpha$, the limit $lim_{xto-infty}alphacdot x=-infty$, the statement remains false, since it is not true for any $alphaleq 0$.

          3. The sentence, applied to another limit, shows how wrong it is to use it to calculate limits. By your logic, $$lim_{xto-infty} frac{1}{x}cdot x=-infty,$$ since we have "something" times "negative infinity". Clearly, this is nonsence - the limit is obviously equal to $1$



          The limit you must calculate is a limit of a product of two numbers, one tends to $0$, the other to $-infty$. Such a limit is often best approached using L'Hospital's rules.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Feb 27 at 10:06

























          answered Feb 27 at 9:52









          5xum5xum

          91.3k394161




          91.3k394161























              2












              $begingroup$

              No. By L'Hopital's Rule $lim frac {frac {pi} 2+arctan, x} {1/x}=lim frac {1/(1+x^{2})} {-1/x^{2}}=-1$.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Yeah I have understood that. That's why I have deleted my comment.
                $endgroup$
                – Dbchatto67
                Feb 27 at 10:18


















              2












              $begingroup$

              No. By L'Hopital's Rule $lim frac {frac {pi} 2+arctan, x} {1/x}=lim frac {1/(1+x^{2})} {-1/x^{2}}=-1$.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Yeah I have understood that. That's why I have deleted my comment.
                $endgroup$
                – Dbchatto67
                Feb 27 at 10:18
















              2












              2








              2





              $begingroup$

              No. By L'Hopital's Rule $lim frac {frac {pi} 2+arctan, x} {1/x}=lim frac {1/(1+x^{2})} {-1/x^{2}}=-1$.






              share|cite|improve this answer









              $endgroup$



              No. By L'Hopital's Rule $lim frac {frac {pi} 2+arctan, x} {1/x}=lim frac {1/(1+x^{2})} {-1/x^{2}}=-1$.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Feb 27 at 9:51









              Kavi Rama MurthyKavi Rama Murthy

              65.1k42766




              65.1k42766












              • $begingroup$
                Yeah I have understood that. That's why I have deleted my comment.
                $endgroup$
                – Dbchatto67
                Feb 27 at 10:18




















              • $begingroup$
                Yeah I have understood that. That's why I have deleted my comment.
                $endgroup$
                – Dbchatto67
                Feb 27 at 10:18


















              $begingroup$
              Yeah I have understood that. That's why I have deleted my comment.
              $endgroup$
              – Dbchatto67
              Feb 27 at 10:18






              $begingroup$
              Yeah I have understood that. That's why I have deleted my comment.
              $endgroup$
              – Dbchatto67
              Feb 27 at 10:18













              0












              $begingroup$

              Let $-1/x=h$



              $$lim_{hto0^+}dfrac{dfracpi2-arctandfrac1h}{-h}=-lim_{...}dfrac{arctan h}h=-1$$



              See Are $mathrm{arccot}(x)$ and $arctan(1/x)$ the same function?






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                Let $-1/x=h$



                $$lim_{hto0^+}dfrac{dfracpi2-arctandfrac1h}{-h}=-lim_{...}dfrac{arctan h}h=-1$$



                See Are $mathrm{arccot}(x)$ and $arctan(1/x)$ the same function?






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  Let $-1/x=h$



                  $$lim_{hto0^+}dfrac{dfracpi2-arctandfrac1h}{-h}=-lim_{...}dfrac{arctan h}h=-1$$



                  See Are $mathrm{arccot}(x)$ and $arctan(1/x)$ the same function?






                  share|cite|improve this answer









                  $endgroup$



                  Let $-1/x=h$



                  $$lim_{hto0^+}dfrac{dfracpi2-arctandfrac1h}{-h}=-lim_{...}dfrac{arctan h}h=-1$$



                  See Are $mathrm{arccot}(x)$ and $arctan(1/x)$ the same function?







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Feb 27 at 15:00









                  lab bhattacharjeelab bhattacharjee

                  226k15157275




                  226k15157275






























                      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%2f3128621%2fdoes-lim-x-to-infty-left-frac-pi2-arctanx-right-cdot-x%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

                      mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924

                      Image
                      0 I just uploaded a new plugin to my wordpress site and I got this warning: mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924 Usually if I have a problem, I can find answers on google but this one seems pretty particular and I'm not finding any info on it except for this forum. I'm not terribly techie but if someone can tell me how to read this and how to change it I can follow directions pretty well. Thanks All! PS If I don't have the proper tags here could someone add the appropriate tags? php mysql wordpress share | improve this question asked Nov 16 '18 at 20:41 ...
                      Read more

                      How to change which sound is reproduced for terminal bell?

                      Image
                      3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question ...
                      Read more

                      Can I use Tabulator js library in my java Spring + Thymeleaf project?

                      Image
                      0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"...
                      Read more