$lim limits_{n to infty} left(1-frac1{n^2}right)^{n}$ equals












2














The limit $lim limits_{n to infty} bigg(1-dfrac{1}{n^2}bigg)^{n}$ equals?



$(A) e^{-1}$



$(B) e^{-frac{1}{2}}$



$(C) e^{-2}$



$(D) 1$



Using the expansion $(1+x)^{n}=1+nx+ dfrac{n(n-1)x^{2}}{2!} ...$

I am having large degree of $n$ in denominator therefor i am left with $1$ after applying the limit. Did i do it right ? If i did it right is there any alternative method to solve this problem like a quick version?






limits exponential-function







share|cite|improve this question





























    2














    The limit $lim limits_{n to infty} bigg(1-dfrac{1}{n^2}bigg)^{n}$ equals?



    $(A) e^{-1}$



    $(B) e^{-frac{1}{2}}$



    $(C) e^{-2}$



    $(D) 1$



    Using the expansion $(1+x)^{n}=1+nx+ dfrac{n(n-1)x^{2}}{2!} ...$

    I am having large degree of $n$ in denominator therefor i am left with $1$ after applying the limit. Did i do it right ? If i did it right is there any alternative method to solve this problem like a quick version?






    limits exponential-function







    share|cite|improve this question



























      2












      2








      2


      1





      The limit $lim limits_{n to infty} bigg(1-dfrac{1}{n^2}bigg)^{n}$ equals?



      $(A) e^{-1}$



      $(B) e^{-frac{1}{2}}$



      $(C) e^{-2}$



      $(D) 1$



      Using the expansion $(1+x)^{n}=1+nx+ dfrac{n(n-1)x^{2}}{2!} ...$

      I am having large degree of $n$ in denominator therefor i am left with $1$ after applying the limit. Did i do it right ? If i did it right is there any alternative method to solve this problem like a quick version?






      limits exponential-function







      share|cite|improve this question















      The limit $lim limits_{n to infty} bigg(1-dfrac{1}{n^2}bigg)^{n}$ equals?



      $(A) e^{-1}$



      $(B) e^{-frac{1}{2}}$



      $(C) e^{-2}$



      $(D) 1$



      Using the expansion $(1+x)^{n}=1+nx+ dfrac{n(n-1)x^{2}}{2!} ...$

      I am having large degree of $n$ in denominator therefor i am left with $1$ after applying the limit. Did i do it right ? If i did it right is there any alternative method to solve this problem like a quick version?






      limits exponential-function




      limits exponential-function






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 9 at 11:03









      Martin Sleziak

      44.7k7115270




      44.7k7115270










      asked Feb 5 at 11:01









      Daman deep

      643218




      643218






















          4 Answers
          4






          active

          oldest

          votes


















          7














          Yes, $lim_{ntoinfty}left(1-frac{1}{n^2}right)^n=lim_{ntoinfty}left(1-frac{1}{n}right)^nleft(1+frac{1}{n}right)^n=lim_{ntoinfty}left(left(1+frac{1}{-n}right)^{-n}right)^{-1}lim_{ntoinfty}left(1+frac{1}{n}right)^n=e^{-1}cdot e=1$






          share|cite|improve this answer





















          • It looks so easy now. How did miss that anyways thanks:)
            – Daman deep
            Feb 5 at 11:12






          • 1




            @Damn1o1 I don't know if you were supposed to solve the problem that way, but at least it serves as a good way to check your result that you may have obtained differently.
            – user491874
            Feb 5 at 11:16










          • No it was best I am building up speed for MCQ exam i shouldn't miss these types of catch. I guess they come with experience.
            – Daman deep
            Feb 5 at 11:18



















          2














          $1-frac{1}{n^2} to 1$ and $n to infty$, so:
          $$lim_{n to infty}left(1-frac{1}{n^2}right)^n=expleft(lim_{n to infty}left(1-frac{1}{n^2}-1right)nright)=expleft(lim_{n to infty}left(-frac{1}{n^2}right)nright)=expleft(lim_{n to infty}-frac{1}{n}right)=exp(-0)=exp(0)=1$$






          share|cite|improve this answer





























            0














            Hint:



            $$lim_{ntoinfty}left(1-dfrac1{n^2}right)^n=left(lim_{ntoinfty}left(1+dfrac1{-n^2}right)^{-n^2}right)^{lim_{ntoinfty}-1/n}$$



            Set $x=1$ in About $lim left(1+frac {x}{n}right)^n$






            share|cite|improve this answer























            • How did you do that with power i don't understand.That makes it $n^3$
              – Daman deep
              Feb 5 at 11:06












            • @Damn1o1, Sorry for the typo
              – lab bhattacharjee
              Feb 5 at 11:07












            • Ohh inside part is $e$ and $e^{-frac{1}{n}}$ makes it 1 yes thank you.
              – Daman deep
              Feb 5 at 11:09





















            0














            By Bernoulli: $ left(1-dfrac1{n^2}right)^n ge 1-frac{1}{n}$, hence



            $$1-frac{1}{n} le left(1-dfrac1{n^2}right)^n le 1.$$



            Conclusion ?






            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',
              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%2f2636871%2flim-limits-n-to-infty-left1-frac1n2-rightn-equals%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              4 Answers
              4






              active

              oldest

              votes








              4 Answers
              4






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              7














              Yes, $lim_{ntoinfty}left(1-frac{1}{n^2}right)^n=lim_{ntoinfty}left(1-frac{1}{n}right)^nleft(1+frac{1}{n}right)^n=lim_{ntoinfty}left(left(1+frac{1}{-n}right)^{-n}right)^{-1}lim_{ntoinfty}left(1+frac{1}{n}right)^n=e^{-1}cdot e=1$






              share|cite|improve this answer





















              • It looks so easy now. How did miss that anyways thanks:)
                – Daman deep
                Feb 5 at 11:12






              • 1




                @Damn1o1 I don't know if you were supposed to solve the problem that way, but at least it serves as a good way to check your result that you may have obtained differently.
                – user491874
                Feb 5 at 11:16










              • No it was best I am building up speed for MCQ exam i shouldn't miss these types of catch. I guess they come with experience.
                – Daman deep
                Feb 5 at 11:18
















              7














              Yes, $lim_{ntoinfty}left(1-frac{1}{n^2}right)^n=lim_{ntoinfty}left(1-frac{1}{n}right)^nleft(1+frac{1}{n}right)^n=lim_{ntoinfty}left(left(1+frac{1}{-n}right)^{-n}right)^{-1}lim_{ntoinfty}left(1+frac{1}{n}right)^n=e^{-1}cdot e=1$






              share|cite|improve this answer





















              • It looks so easy now. How did miss that anyways thanks:)
                – Daman deep
                Feb 5 at 11:12






              • 1




                @Damn1o1 I don't know if you were supposed to solve the problem that way, but at least it serves as a good way to check your result that you may have obtained differently.
                – user491874
                Feb 5 at 11:16










              • No it was best I am building up speed for MCQ exam i shouldn't miss these types of catch. I guess they come with experience.
                – Daman deep
                Feb 5 at 11:18














              7












              7








              7






              Yes, $lim_{ntoinfty}left(1-frac{1}{n^2}right)^n=lim_{ntoinfty}left(1-frac{1}{n}right)^nleft(1+frac{1}{n}right)^n=lim_{ntoinfty}left(left(1+frac{1}{-n}right)^{-n}right)^{-1}lim_{ntoinfty}left(1+frac{1}{n}right)^n=e^{-1}cdot e=1$






              share|cite|improve this answer












              Yes, $lim_{ntoinfty}left(1-frac{1}{n^2}right)^n=lim_{ntoinfty}left(1-frac{1}{n}right)^nleft(1+frac{1}{n}right)^n=lim_{ntoinfty}left(left(1+frac{1}{-n}right)^{-n}right)^{-1}lim_{ntoinfty}left(1+frac{1}{n}right)^n=e^{-1}cdot e=1$







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Feb 5 at 11:05







              user491874



















              • It looks so easy now. How did miss that anyways thanks:)
                – Daman deep
                Feb 5 at 11:12






              • 1




                @Damn1o1 I don't know if you were supposed to solve the problem that way, but at least it serves as a good way to check your result that you may have obtained differently.
                – user491874
                Feb 5 at 11:16










              • No it was best I am building up speed for MCQ exam i shouldn't miss these types of catch. I guess they come with experience.
                – Daman deep
                Feb 5 at 11:18


















              • It looks so easy now. How did miss that anyways thanks:)
                – Daman deep
                Feb 5 at 11:12






              • 1




                @Damn1o1 I don't know if you were supposed to solve the problem that way, but at least it serves as a good way to check your result that you may have obtained differently.
                – user491874
                Feb 5 at 11:16










              • No it was best I am building up speed for MCQ exam i shouldn't miss these types of catch. I guess they come with experience.
                – Daman deep
                Feb 5 at 11:18
















              It looks so easy now. How did miss that anyways thanks:)
              – Daman deep
              Feb 5 at 11:12




              It looks so easy now. How did miss that anyways thanks:)
              – Daman deep
              Feb 5 at 11:12




              1




              1




              @Damn1o1 I don't know if you were supposed to solve the problem that way, but at least it serves as a good way to check your result that you may have obtained differently.
              – user491874
              Feb 5 at 11:16




              @Damn1o1 I don't know if you were supposed to solve the problem that way, but at least it serves as a good way to check your result that you may have obtained differently.
              – user491874
              Feb 5 at 11:16












              No it was best I am building up speed for MCQ exam i shouldn't miss these types of catch. I guess they come with experience.
              – Daman deep
              Feb 5 at 11:18




              No it was best I am building up speed for MCQ exam i shouldn't miss these types of catch. I guess they come with experience.
              – Daman deep
              Feb 5 at 11:18











              2














              $1-frac{1}{n^2} to 1$ and $n to infty$, so:
              $$lim_{n to infty}left(1-frac{1}{n^2}right)^n=expleft(lim_{n to infty}left(1-frac{1}{n^2}-1right)nright)=expleft(lim_{n to infty}left(-frac{1}{n^2}right)nright)=expleft(lim_{n to infty}-frac{1}{n}right)=exp(-0)=exp(0)=1$$






              share|cite|improve this answer


























                2














                $1-frac{1}{n^2} to 1$ and $n to infty$, so:
                $$lim_{n to infty}left(1-frac{1}{n^2}right)^n=expleft(lim_{n to infty}left(1-frac{1}{n^2}-1right)nright)=expleft(lim_{n to infty}left(-frac{1}{n^2}right)nright)=expleft(lim_{n to infty}-frac{1}{n}right)=exp(-0)=exp(0)=1$$






                share|cite|improve this answer
























                  2












                  2








                  2






                  $1-frac{1}{n^2} to 1$ and $n to infty$, so:
                  $$lim_{n to infty}left(1-frac{1}{n^2}right)^n=expleft(lim_{n to infty}left(1-frac{1}{n^2}-1right)nright)=expleft(lim_{n to infty}left(-frac{1}{n^2}right)nright)=expleft(lim_{n to infty}-frac{1}{n}right)=exp(-0)=exp(0)=1$$






                  share|cite|improve this answer












                  $1-frac{1}{n^2} to 1$ and $n to infty$, so:
                  $$lim_{n to infty}left(1-frac{1}{n^2}right)^n=expleft(lim_{n to infty}left(1-frac{1}{n^2}-1right)nright)=expleft(lim_{n to infty}left(-frac{1}{n^2}right)nright)=expleft(lim_{n to infty}-frac{1}{n}right)=exp(-0)=exp(0)=1$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Feb 5 at 11:20









                  Botond

                  5,5232732




                  5,5232732























                      0














                      Hint:



                      $$lim_{ntoinfty}left(1-dfrac1{n^2}right)^n=left(lim_{ntoinfty}left(1+dfrac1{-n^2}right)^{-n^2}right)^{lim_{ntoinfty}-1/n}$$



                      Set $x=1$ in About $lim left(1+frac {x}{n}right)^n$






                      share|cite|improve this answer























                      • How did you do that with power i don't understand.That makes it $n^3$
                        – Daman deep
                        Feb 5 at 11:06












                      • @Damn1o1, Sorry for the typo
                        – lab bhattacharjee
                        Feb 5 at 11:07












                      • Ohh inside part is $e$ and $e^{-frac{1}{n}}$ makes it 1 yes thank you.
                        – Daman deep
                        Feb 5 at 11:09


















                      0














                      Hint:



                      $$lim_{ntoinfty}left(1-dfrac1{n^2}right)^n=left(lim_{ntoinfty}left(1+dfrac1{-n^2}right)^{-n^2}right)^{lim_{ntoinfty}-1/n}$$



                      Set $x=1$ in About $lim left(1+frac {x}{n}right)^n$






                      share|cite|improve this answer























                      • How did you do that with power i don't understand.That makes it $n^3$
                        – Daman deep
                        Feb 5 at 11:06












                      • @Damn1o1, Sorry for the typo
                        – lab bhattacharjee
                        Feb 5 at 11:07












                      • Ohh inside part is $e$ and $e^{-frac{1}{n}}$ makes it 1 yes thank you.
                        – Daman deep
                        Feb 5 at 11:09
















                      0












                      0








                      0






                      Hint:



                      $$lim_{ntoinfty}left(1-dfrac1{n^2}right)^n=left(lim_{ntoinfty}left(1+dfrac1{-n^2}right)^{-n^2}right)^{lim_{ntoinfty}-1/n}$$



                      Set $x=1$ in About $lim left(1+frac {x}{n}right)^n$






                      share|cite|improve this answer














                      Hint:



                      $$lim_{ntoinfty}left(1-dfrac1{n^2}right)^n=left(lim_{ntoinfty}left(1+dfrac1{-n^2}right)^{-n^2}right)^{lim_{ntoinfty}-1/n}$$



                      Set $x=1$ in About $lim left(1+frac {x}{n}right)^n$







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited Feb 5 at 11:07

























                      answered Feb 5 at 11:03









                      lab bhattacharjee

                      223k15156274




                      223k15156274












                      • How did you do that with power i don't understand.That makes it $n^3$
                        – Daman deep
                        Feb 5 at 11:06












                      • @Damn1o1, Sorry for the typo
                        – lab bhattacharjee
                        Feb 5 at 11:07












                      • Ohh inside part is $e$ and $e^{-frac{1}{n}}$ makes it 1 yes thank you.
                        – Daman deep
                        Feb 5 at 11:09




















                      • How did you do that with power i don't understand.That makes it $n^3$
                        – Daman deep
                        Feb 5 at 11:06












                      • @Damn1o1, Sorry for the typo
                        – lab bhattacharjee
                        Feb 5 at 11:07












                      • Ohh inside part is $e$ and $e^{-frac{1}{n}}$ makes it 1 yes thank you.
                        – Daman deep
                        Feb 5 at 11:09


















                      How did you do that with power i don't understand.That makes it $n^3$
                      – Daman deep
                      Feb 5 at 11:06






                      How did you do that with power i don't understand.That makes it $n^3$
                      – Daman deep
                      Feb 5 at 11:06














                      @Damn1o1, Sorry for the typo
                      – lab bhattacharjee
                      Feb 5 at 11:07






                      @Damn1o1, Sorry for the typo
                      – lab bhattacharjee
                      Feb 5 at 11:07














                      Ohh inside part is $e$ and $e^{-frac{1}{n}}$ makes it 1 yes thank you.
                      – Daman deep
                      Feb 5 at 11:09






                      Ohh inside part is $e$ and $e^{-frac{1}{n}}$ makes it 1 yes thank you.
                      – Daman deep
                      Feb 5 at 11:09













                      0














                      By Bernoulli: $ left(1-dfrac1{n^2}right)^n ge 1-frac{1}{n}$, hence



                      $$1-frac{1}{n} le left(1-dfrac1{n^2}right)^n le 1.$$



                      Conclusion ?






                      share|cite|improve this answer


























                        0














                        By Bernoulli: $ left(1-dfrac1{n^2}right)^n ge 1-frac{1}{n}$, hence



                        $$1-frac{1}{n} le left(1-dfrac1{n^2}right)^n le 1.$$



                        Conclusion ?






                        share|cite|improve this answer
























                          0












                          0








                          0






                          By Bernoulli: $ left(1-dfrac1{n^2}right)^n ge 1-frac{1}{n}$, hence



                          $$1-frac{1}{n} le left(1-dfrac1{n^2}right)^n le 1.$$



                          Conclusion ?






                          share|cite|improve this answer












                          By Bernoulli: $ left(1-dfrac1{n^2}right)^n ge 1-frac{1}{n}$, hence



                          $$1-frac{1}{n} le left(1-dfrac1{n^2}right)^n le 1.$$



                          Conclusion ?







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Feb 5 at 12:04









                          Fred

                          44.2k1845




                          44.2k1845






























                              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.





                              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.




                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function () {
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2636871%2flim-limits-n-to-infty-left1-frac1n2-rightn-equals%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

                              Bressuire

                              Image
                              .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid
                              Read more

                              Cabo Verde

                              Image
                                Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º)
                              Read more

                              Gyllenstierna

                              Image
                              .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid
                              Read more