Limit using l'Hopital's rule with logaritmus












0















What is $lim_{xrightarrow0}(1+2sin x)^frac{1}{tan x}$?




Computing $lim_{xrightarrow0}log(1+2sin x)^frac{1}{tan x}=lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}$



Here, where I can use l'Hopital's rule I get:



$lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}=2$



But now my question is:



$lim_{xrightarrow0}(1+2sin x)^frac{1}{tan x}$ = $lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}=2$?



I don't think they are the same, but can I remove the log? Maybe it's easier than I think, but now I don't know how to go on.






real-analysis limits







share|cite|improve this question





























    0















    What is $lim_{xrightarrow0}(1+2sin x)^frac{1}{tan x}$?




    Computing $lim_{xrightarrow0}log(1+2sin x)^frac{1}{tan x}=lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}$



    Here, where I can use l'Hopital's rule I get:



    $lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}=2$



    But now my question is:



    $lim_{xrightarrow0}(1+2sin x)^frac{1}{tan x}$ = $lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}=2$?



    I don't think they are the same, but can I remove the log? Maybe it's easier than I think, but now I don't know how to go on.






    real-analysis limits







    share|cite|improve this question



























      0












      0








      0








      What is $lim_{xrightarrow0}(1+2sin x)^frac{1}{tan x}$?




      Computing $lim_{xrightarrow0}log(1+2sin x)^frac{1}{tan x}=lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}$



      Here, where I can use l'Hopital's rule I get:



      $lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}=2$



      But now my question is:



      $lim_{xrightarrow0}(1+2sin x)^frac{1}{tan x}$ = $lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}=2$?



      I don't think they are the same, but can I remove the log? Maybe it's easier than I think, but now I don't know how to go on.






      real-analysis limits







      share|cite|improve this question
















      What is $lim_{xrightarrow0}(1+2sin x)^frac{1}{tan x}$?




      Computing $lim_{xrightarrow0}log(1+2sin x)^frac{1}{tan x}=lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}$



      Here, where I can use l'Hopital's rule I get:



      $lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}=2$



      But now my question is:



      $lim_{xrightarrow0}(1+2sin x)^frac{1}{tan x}$ = $lim_{xrightarrow0}frac{log(1+2sin x)}{tan x}=2$?



      I don't think they are the same, but can I remove the log? Maybe it's easier than I think, but now I don't know how to go on.






      real-analysis limits




      real-analysis limits






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 29 at 22:31









      gimusi

      1




      1










      asked Nov 29 at 22:19









      Dada

      7010




      7010






















          3 Answers
          3






          active

          oldest

          votes


















          2














          You mean $$lim_{xto 0}(1+2sin x)^{tfrac{1}{tan x}}=explim_{xto 0}frac{ln (1+2sin x)}{tan x}=e^2.$$






          share|cite|improve this answer





























            2














            The first equality you write is incorrect. You can evaluate
            $$
            lim_{xto0}frac{log(1+2sin x)}{tan x}
            $$
            in order to compute the given limit, but they are not equal; if $l$ is the latter limit, then the one you're looking for is $e^l$.



            Now, with l'Hôpital,
            $$
            lim_{xto0}frac{log(1+2sin x)}{tan x}=
            lim_{xto0}frac{dfrac{2cos x}{1+2sin x}}{1+tan^2x}=2
            $$
            and therefore
            $$
            lim_{xto0}(1+2sin x)^{1/tan x}=e^2
            $$






            share|cite|improve this answer





























              2














              As an alternative



              $$large (1+2 sin x)^frac{1}{tan x}=left[(1+2 sin x)^{frac1{2sin x}}right]^frac{2sin x}{tan x} to e^2$$






              share|cite|improve this answer





















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                3 Answers
                3






                active

                oldest

                votes








                3 Answers
                3






                active

                oldest

                votes









                active

                oldest

                votes






                active

                oldest

                votes









                2














                You mean $$lim_{xto 0}(1+2sin x)^{tfrac{1}{tan x}}=explim_{xto 0}frac{ln (1+2sin x)}{tan x}=e^2.$$






                share|cite|improve this answer


























                  2














                  You mean $$lim_{xto 0}(1+2sin x)^{tfrac{1}{tan x}}=explim_{xto 0}frac{ln (1+2sin x)}{tan x}=e^2.$$






                  share|cite|improve this answer
























                    2












                    2








                    2






                    You mean $$lim_{xto 0}(1+2sin x)^{tfrac{1}{tan x}}=explim_{xto 0}frac{ln (1+2sin x)}{tan x}=e^2.$$






                    share|cite|improve this answer












                    You mean $$lim_{xto 0}(1+2sin x)^{tfrac{1}{tan x}}=explim_{xto 0}frac{ln (1+2sin x)}{tan x}=e^2.$$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Nov 29 at 22:26









                    J.G.

                    22.3k22034




                    22.3k22034























                        2














                        The first equality you write is incorrect. You can evaluate
                        $$
                        lim_{xto0}frac{log(1+2sin x)}{tan x}
                        $$
                        in order to compute the given limit, but they are not equal; if $l$ is the latter limit, then the one you're looking for is $e^l$.



                        Now, with l'Hôpital,
                        $$
                        lim_{xto0}frac{log(1+2sin x)}{tan x}=
                        lim_{xto0}frac{dfrac{2cos x}{1+2sin x}}{1+tan^2x}=2
                        $$
                        and therefore
                        $$
                        lim_{xto0}(1+2sin x)^{1/tan x}=e^2
                        $$






                        share|cite|improve this answer


























                          2














                          The first equality you write is incorrect. You can evaluate
                          $$
                          lim_{xto0}frac{log(1+2sin x)}{tan x}
                          $$
                          in order to compute the given limit, but they are not equal; if $l$ is the latter limit, then the one you're looking for is $e^l$.



                          Now, with l'Hôpital,
                          $$
                          lim_{xto0}frac{log(1+2sin x)}{tan x}=
                          lim_{xto0}frac{dfrac{2cos x}{1+2sin x}}{1+tan^2x}=2
                          $$
                          and therefore
                          $$
                          lim_{xto0}(1+2sin x)^{1/tan x}=e^2
                          $$






                          share|cite|improve this answer
























                            2












                            2








                            2






                            The first equality you write is incorrect. You can evaluate
                            $$
                            lim_{xto0}frac{log(1+2sin x)}{tan x}
                            $$
                            in order to compute the given limit, but they are not equal; if $l$ is the latter limit, then the one you're looking for is $e^l$.



                            Now, with l'Hôpital,
                            $$
                            lim_{xto0}frac{log(1+2sin x)}{tan x}=
                            lim_{xto0}frac{dfrac{2cos x}{1+2sin x}}{1+tan^2x}=2
                            $$
                            and therefore
                            $$
                            lim_{xto0}(1+2sin x)^{1/tan x}=e^2
                            $$






                            share|cite|improve this answer












                            The first equality you write is incorrect. You can evaluate
                            $$
                            lim_{xto0}frac{log(1+2sin x)}{tan x}
                            $$
                            in order to compute the given limit, but they are not equal; if $l$ is the latter limit, then the one you're looking for is $e^l$.



                            Now, with l'Hôpital,
                            $$
                            lim_{xto0}frac{log(1+2sin x)}{tan x}=
                            lim_{xto0}frac{dfrac{2cos x}{1+2sin x}}{1+tan^2x}=2
                            $$
                            and therefore
                            $$
                            lim_{xto0}(1+2sin x)^{1/tan x}=e^2
                            $$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Nov 29 at 22:26









                            egreg

                            177k1484200




                            177k1484200























                                2














                                As an alternative



                                $$large (1+2 sin x)^frac{1}{tan x}=left[(1+2 sin x)^{frac1{2sin x}}right]^frac{2sin x}{tan x} to e^2$$






                                share|cite|improve this answer


























                                  2














                                  As an alternative



                                  $$large (1+2 sin x)^frac{1}{tan x}=left[(1+2 sin x)^{frac1{2sin x}}right]^frac{2sin x}{tan x} to e^2$$






                                  share|cite|improve this answer
























                                    2












                                    2








                                    2






                                    As an alternative



                                    $$large (1+2 sin x)^frac{1}{tan x}=left[(1+2 sin x)^{frac1{2sin x}}right]^frac{2sin x}{tan x} to e^2$$






                                    share|cite|improve this answer












                                    As an alternative



                                    $$large (1+2 sin x)^frac{1}{tan x}=left[(1+2 sin x)^{frac1{2sin x}}right]^frac{2sin x}{tan x} to e^2$$







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered Nov 29 at 22:28









                                    gimusi

                                    1




                                    1






























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