How do I solve this limit? The Next CEO of Stack Overflowrational limit problemHow to evaluate the following limit? $limlimits_xtoinftyxleft(fracpi2-arctan xright).$Limit only using squeeze theorem $ lim_(x,y)to(0,2) x,arctanleft(frac1y-2right)$How to calculate this limit at infinity?How can I use the Limit Laws to solve this limit?How do I even start approaching this limit?how can I find this limitHow to solve this limit without L'Hospital?How to solve this limit when direct substitution fails. Why do this work?L'Hôpital's rule - How solve this limit question

Implement the Thanos sorting algorithm

too much space between section and text in a twocolumn document

How do I solve this limit?

Increase performance creating Mandelbrot set in python

Can a single photon have an energy density?

Inappropriate reference requests from Journal reviewers

Grabbing quick drinks

What is the difference between "behavior" and "behaviour"?

What is the purpose of the Evocation wizard's Potent Cantrip feature?

What is meant by a M next to a roman numeral?

declare as function pointer and initialize in the same line

How to use tikz in fbox?

% symbol leads to superlong (forever?) compilations

Is HostGator storing my password in plaintext?

Rotate a column

Why do professional authors make "consistency" mistakes? And how to avoid them?

Are there languages with no euphemisms?

How to write papers efficiently when English isn't my first language?

Would this house-rule that treats advantage as a +1 to the roll instead (and disadvantage as -1) and allows them to stack be balanced?

How to write the block matrix in LaTex?

How long to clear the 'suck zone' of a turbofan after start is initiated?

Opposite of a diet

Why didn't Theresa May consult with Parliament before negotiating a deal with the EU?

How do scammers retract money, while you can’t?



How do I solve this limit?



The Next CEO of Stack Overflowrational limit problemHow to evaluate the following limit? $limlimits_xtoinftyxleft(fracpi2-arctan xright).$Limit only using squeeze theorem $ lim_(x,y)to(0,2) x,arctanleft(frac1y-2right)$How to calculate this limit at infinity?How can I use the Limit Laws to solve this limit?How do I even start approaching this limit?how can I find this limitHow to solve this limit without L'Hospital?How to solve this limit when direct substitution fails. Why do this work?L'Hôpital's rule - How solve this limit question










1












$begingroup$


How do I solve:



$$
lim_x to 0left[1 - xarctanleft(nxright)right]^, 1/x^2
$$



I know the answer is $e^-n$ for every n > 1, but for the life of me I have no idea how to actually get to that answer. Am I supposed to use a common limit or any theorem? Also, why isn't the limit equal to 1?



Thanks.






calculus limits







share|cite|improve this question









New contributor




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







$endgroup$











  • $begingroup$
    I accidentally put k instead of n, sorry.
    $endgroup$
    – radoo
    1 hour ago










  • $begingroup$
    You can take the logarithm of the function and use L'Hopitals rule
    $endgroup$
    – Nimish
    1 hour ago















1












$begingroup$


How do I solve:



$$
lim_x to 0left[1 - xarctanleft(nxright)right]^, 1/x^2
$$



I know the answer is $e^-n$ for every n > 1, but for the life of me I have no idea how to actually get to that answer. Am I supposed to use a common limit or any theorem? Also, why isn't the limit equal to 1?



Thanks.






calculus limits







share|cite|improve this question









New contributor




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







$endgroup$











  • $begingroup$
    I accidentally put k instead of n, sorry.
    $endgroup$
    – radoo
    1 hour ago










  • $begingroup$
    You can take the logarithm of the function and use L'Hopitals rule
    $endgroup$
    – Nimish
    1 hour ago













1












1








1





$begingroup$


How do I solve:



$$
lim_x to 0left[1 - xarctanleft(nxright)right]^, 1/x^2
$$



I know the answer is $e^-n$ for every n > 1, but for the life of me I have no idea how to actually get to that answer. Am I supposed to use a common limit or any theorem? Also, why isn't the limit equal to 1?



Thanks.






calculus limits







share|cite|improve this question









New contributor




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







$endgroup$




How do I solve:



$$
lim_x to 0left[1 - xarctanleft(nxright)right]^, 1/x^2
$$



I know the answer is $e^-n$ for every n > 1, but for the life of me I have no idea how to actually get to that answer. Am I supposed to use a common limit or any theorem? Also, why isn't the limit equal to 1?



Thanks.






calculus limits




calculus limits






share|cite|improve this question









New contributor




radoo 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 question









New contributor




radoo 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 question




share|cite|improve this question








edited 1 hour ago









Felix Marin

68.8k7109146




68.8k7109146






New contributor




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









asked 2 hours ago









radooradoo

84




84




New contributor




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





New contributor





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






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











  • $begingroup$
    I accidentally put k instead of n, sorry.
    $endgroup$
    – radoo
    1 hour ago










  • $begingroup$
    You can take the logarithm of the function and use L'Hopitals rule
    $endgroup$
    – Nimish
    1 hour ago
















  • $begingroup$
    I accidentally put k instead of n, sorry.
    $endgroup$
    – radoo
    1 hour ago










  • $begingroup$
    You can take the logarithm of the function and use L'Hopitals rule
    $endgroup$
    – Nimish
    1 hour ago















$begingroup$
I accidentally put k instead of n, sorry.
$endgroup$
– radoo
1 hour ago




$begingroup$
I accidentally put k instead of n, sorry.
$endgroup$
– radoo
1 hour ago












$begingroup$
You can take the logarithm of the function and use L'Hopitals rule
$endgroup$
– Nimish
1 hour ago




$begingroup$
You can take the logarithm of the function and use L'Hopitals rule
$endgroup$
– Nimish
1 hour ago










3 Answers
3






active

oldest

votes


















3












$begingroup$

You can do it the following way:



$$
lim_xto 0 (1-xarctan(nx))^1/x^2 = lim_xto 0 e^log((1-xarctan(nx))^1/x^2)
$$



doing some algebra on the exponent:
$$
lim_xto 0 exp(log((1-xarctan(nx))^1/x^2)) = lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right)
$$



by limit rules:



$$
lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right) =expleft[ lim_xto 0 left(fraclog((1-xarctan(nx))x^2)right)right]
$$



apply L'Hospital's rule and after some algebra you should get:
$$
expleft[- fracn+lim_xto 0 fracarctan(nx)x+n^2lim_xto 0 xarctan(nx)2right]
$$



simplifying



$$
expleft(-fracn+n+n^22right) = e^-n
$$






share|cite|improve this answer









$endgroup$




















    2












    $begingroup$

    Hint: $lim_xto 0 (1-nx)^(1/x)=e^-n$



    Also look at the Taylor expansion of the arctan






    share|cite|improve this answer









    $endgroup$




















      0












      $begingroup$

      Noting that $u=xarctan nxto0$ and $(arctan nx)/xto n$ as $xto 0$, we have



      $$(1-xarctan nx)^1/x^2=((1-xarctan nx)^1/(xarctan nx))^(arctan nx)/x=((1-u)^1/u)^(arctan nx)/xto (e^-1)^n=e^-n$$



      using the general limit property $lim f(x)^g(x)=(lim f(x))^lim g(x)$, provided $lim f(x)$ and $lim g(x)$ both exist (with $lim f(x)ge0$) and are not both $0$.






      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
        );



        );






        radoo is a new contributor. Be nice, and check out our Code of Conduct.









        draft saved

        draft discarded


















        StackExchange.ready(
        function ()
        StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3166364%2fhow-do-i-solve-this-limit%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









        3












        $begingroup$

        You can do it the following way:



        $$
        lim_xto 0 (1-xarctan(nx))^1/x^2 = lim_xto 0 e^log((1-xarctan(nx))^1/x^2)
        $$



        doing some algebra on the exponent:
        $$
        lim_xto 0 exp(log((1-xarctan(nx))^1/x^2)) = lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right)
        $$



        by limit rules:



        $$
        lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right) =expleft[ lim_xto 0 left(fraclog((1-xarctan(nx))x^2)right)right]
        $$



        apply L'Hospital's rule and after some algebra you should get:
        $$
        expleft[- fracn+lim_xto 0 fracarctan(nx)x+n^2lim_xto 0 xarctan(nx)2right]
        $$



        simplifying



        $$
        expleft(-fracn+n+n^22right) = e^-n
        $$






        share|cite|improve this answer









        $endgroup$

















          3












          $begingroup$

          You can do it the following way:



          $$
          lim_xto 0 (1-xarctan(nx))^1/x^2 = lim_xto 0 e^log((1-xarctan(nx))^1/x^2)
          $$



          doing some algebra on the exponent:
          $$
          lim_xto 0 exp(log((1-xarctan(nx))^1/x^2)) = lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right)
          $$



          by limit rules:



          $$
          lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right) =expleft[ lim_xto 0 left(fraclog((1-xarctan(nx))x^2)right)right]
          $$



          apply L'Hospital's rule and after some algebra you should get:
          $$
          expleft[- fracn+lim_xto 0 fracarctan(nx)x+n^2lim_xto 0 xarctan(nx)2right]
          $$



          simplifying



          $$
          expleft(-fracn+n+n^22right) = e^-n
          $$






          share|cite|improve this answer









          $endgroup$















            3












            3








            3





            $begingroup$

            You can do it the following way:



            $$
            lim_xto 0 (1-xarctan(nx))^1/x^2 = lim_xto 0 e^log((1-xarctan(nx))^1/x^2)
            $$



            doing some algebra on the exponent:
            $$
            lim_xto 0 exp(log((1-xarctan(nx))^1/x^2)) = lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right)
            $$



            by limit rules:



            $$
            lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right) =expleft[ lim_xto 0 left(fraclog((1-xarctan(nx))x^2)right)right]
            $$



            apply L'Hospital's rule and after some algebra you should get:
            $$
            expleft[- fracn+lim_xto 0 fracarctan(nx)x+n^2lim_xto 0 xarctan(nx)2right]
            $$



            simplifying



            $$
            expleft(-fracn+n+n^22right) = e^-n
            $$






            share|cite|improve this answer









            $endgroup$



            You can do it the following way:



            $$
            lim_xto 0 (1-xarctan(nx))^1/x^2 = lim_xto 0 e^log((1-xarctan(nx))^1/x^2)
            $$



            doing some algebra on the exponent:
            $$
            lim_xto 0 exp(log((1-xarctan(nx))^1/x^2)) = lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right)
            $$



            by limit rules:



            $$
            lim_xto 0 expleft(fraclog((1-xarctan(nx))x^2)right) =expleft[ lim_xto 0 left(fraclog((1-xarctan(nx))x^2)right)right]
            $$



            apply L'Hospital's rule and after some algebra you should get:
            $$
            expleft[- fracn+lim_xto 0 fracarctan(nx)x+n^2lim_xto 0 xarctan(nx)2right]
            $$



            simplifying



            $$
            expleft(-fracn+n+n^22right) = e^-n
            $$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 1 hour ago









            DashiDashi

            726311




            726311





















                2












                $begingroup$

                Hint: $lim_xto 0 (1-nx)^(1/x)=e^-n$



                Also look at the Taylor expansion of the arctan






                share|cite|improve this answer









                $endgroup$

















                  2












                  $begingroup$

                  Hint: $lim_xto 0 (1-nx)^(1/x)=e^-n$



                  Also look at the Taylor expansion of the arctan






                  share|cite|improve this answer









                  $endgroup$















                    2












                    2








                    2





                    $begingroup$

                    Hint: $lim_xto 0 (1-nx)^(1/x)=e^-n$



                    Also look at the Taylor expansion of the arctan






                    share|cite|improve this answer









                    $endgroup$



                    Hint: $lim_xto 0 (1-nx)^(1/x)=e^-n$



                    Also look at the Taylor expansion of the arctan







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 1 hour ago









                    A. PA. P

                    1386




                    1386





















                        0












                        $begingroup$

                        Noting that $u=xarctan nxto0$ and $(arctan nx)/xto n$ as $xto 0$, we have



                        $$(1-xarctan nx)^1/x^2=((1-xarctan nx)^1/(xarctan nx))^(arctan nx)/x=((1-u)^1/u)^(arctan nx)/xto (e^-1)^n=e^-n$$



                        using the general limit property $lim f(x)^g(x)=(lim f(x))^lim g(x)$, provided $lim f(x)$ and $lim g(x)$ both exist (with $lim f(x)ge0$) and are not both $0$.






                        share|cite|improve this answer









                        $endgroup$

















                          0












                          $begingroup$

                          Noting that $u=xarctan nxto0$ and $(arctan nx)/xto n$ as $xto 0$, we have



                          $$(1-xarctan nx)^1/x^2=((1-xarctan nx)^1/(xarctan nx))^(arctan nx)/x=((1-u)^1/u)^(arctan nx)/xto (e^-1)^n=e^-n$$



                          using the general limit property $lim f(x)^g(x)=(lim f(x))^lim g(x)$, provided $lim f(x)$ and $lim g(x)$ both exist (with $lim f(x)ge0$) and are not both $0$.






                          share|cite|improve this answer









                          $endgroup$















                            0












                            0








                            0





                            $begingroup$

                            Noting that $u=xarctan nxto0$ and $(arctan nx)/xto n$ as $xto 0$, we have



                            $$(1-xarctan nx)^1/x^2=((1-xarctan nx)^1/(xarctan nx))^(arctan nx)/x=((1-u)^1/u)^(arctan nx)/xto (e^-1)^n=e^-n$$



                            using the general limit property $lim f(x)^g(x)=(lim f(x))^lim g(x)$, provided $lim f(x)$ and $lim g(x)$ both exist (with $lim f(x)ge0$) and are not both $0$.






                            share|cite|improve this answer









                            $endgroup$



                            Noting that $u=xarctan nxto0$ and $(arctan nx)/xto n$ as $xto 0$, we have



                            $$(1-xarctan nx)^1/x^2=((1-xarctan nx)^1/(xarctan nx))^(arctan nx)/x=((1-u)^1/u)^(arctan nx)/xto (e^-1)^n=e^-n$$



                            using the general limit property $lim f(x)^g(x)=(lim f(x))^lim g(x)$, provided $lim f(x)$ and $lim g(x)$ both exist (with $lim f(x)ge0$) and are not both $0$.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 34 mins ago









                            Barry CipraBarry Cipra

                            60.5k655128




                            60.5k655128




















                                radoo is a new contributor. Be nice, and check out our Code of Conduct.









                                draft saved

                                draft discarded


















                                radoo is a new contributor. Be nice, and check out our Code of Conduct.












                                radoo is a new contributor. Be nice, and check out our Code of Conduct.











                                radoo is a new contributor. Be nice, and check out our Code of Conduct.














                                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%2f3166364%2fhow-do-i-solve-this-limit%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

                                Is it wise to focus on putting odd beats on left when playing double bass drums?What does the tie across other different notes mean?How to improve right hand picking/strumming technique for electric and acoustic guitar?How fast is the BPM should you start to use double bass pedal?Alternative or more advanced methods for counting rhythmsLost direction in practicing drumsIs there any way at all to tell if these measures are 6/4 vs. 3/2?Why is alberti bass so tiring?How does 'meter' differ from 'rhythm' in music?Rhythm question about piano sheet notationWhy is Bach's Cello Suite 1 written in 16th notes

                                Image
                                Is there a familial term for apples and pears? Are white and non-white police officers equally likely to kill black suspects? How would photo IDs work for shapeshifters? Denied boarding due to overcrowding, Sparpreis ticket. What are my rights? Crop image to path created in TikZ? extract characters between two commas? What are the advantages and disadvantages of running one shots compared to campaigns? Add an angle to a sphere Can I legally use front facing blue light in the UK? Is there a way to make member function NOT callable from constructor? What does "enim et" mean? I’m planning on buying a laser printer but concerned about the life cycle of toner in the machine How did the USSR manage to innovate in an environment characterized by government censorship and high bureaucracy? I see my dog run How is it possible for user's password to be changed after storage was encrypted? (on OS X, Android) What is the offset in a seaplane's hull? ...
                                Read more

                                History of India Contents Prehistoric era (until c. 3300 BCE) Bronze Age – "First urbanisation" (c. 3300 – c. 1800 BCE) Climate change, de-urbanisation, and Indo-Aryan migrations (c.1800 – 1500 BCE) Iron Age - Vedic period (c. 1500 – c. 600 BCE) "Second urbanisation" (c. 600 – c. 200 BCE) Classical to early medieval periods (c. 200 BCE – c. 1200 CE) Late medieval period (c. 1200 – 1526 CE) Early modern period (c. 1526–1858 CE) Modern period and independence (after c. 1850 CE) Historiography See also References Further reading External links Navigation menureviewedee[update]The crisisThe Evolution and History of Human Populations in South Asia: Inter-disciplinary Studies in Archaeology, Biological Anthropology, Linguistics and GeneticsThe Ancient Indus: Urbanism, Economy, and Society"Indus River Valley Civilizations"The Ancient Indus: Urbanism, Economy, and Society"India before the British: The Mughal Empire and its Rivals, 1526–1857"Why Europe Grew Rich and Asia Did Not: Global Economic Divergence, 1600–1850Development Centre Studies The World Economy Historical Statistics: Historical StatisticsDeveloping cultures: case studiesEthnic Groups of South Asia and the Pacific: An Encyclopedia: An Encyclopedia"Indian Economy During British Rule""Economic Impact of the British Rule in India | Indian History"The Bone Readers: Science and Politics in Human Origins Research"Out of Africa: new hypotheses and evidence for the dispersal of Homo sapiens along the Indian Ocean rim"10.3109/0301446100363924920334598"Genetic and archaeological perspectives on the initial modern human colonization of southern Asia"2013PNAS..11010699M10.1073/pnas.1306043110369678523754394"Edakkal Caves|Places Around in Wayanad"Protecting megaliths to keep history alive The Hindu daily"Archaeologists rock solid behind Edakkal Cave"The Global Prehistory of Human Migration"Indus Valley 2,000 years older than thought"the original"Stepwells -Cosmology of Subterranean Architecture as seen in Adalaj"A History of Ancient and Early medieval India : from the Stone Age to the 12th century"Stone celts in Harappa"the original"Peoples and languages in pre-islamic Indus valley"the original"The Sindhi language"the originalThe Aryan chromosome"Fluvial landscapes of the Harappan Civilization"2012PNAS..109E1688G10.1073/pnas.1112743109338705422645375"Is River Ghaggar, Saraswati? Geochemical Constraints""An Ancient Civilization, Upended by Climate Change""Huge Ancient Civilization's Collapse Explained"2003GeoRL..30.1425S10.1029/2002GL0168222006QSRv...25.1283M10.1016/j.quascirev.2005.10.0122011QuInt.229..140M10.1016/j.quaint.2009.11.012Climate Change and the Course of Global History: A Rough JourneyA History of Ancient and Early Medieval India: From the Stone Age to the 12th CenturyA History of Ancient and Mediaeval India: From the Stone Age to the 12th CenturyAntennae SwordA History of IndiaA History of IndiaAn Introduction to Hinduism"India: The Late 2nd Millennium and the Reemergence of Urbanism"The Coinage of Ancient IndiaA Sanskrit reader: with vocabulary and notesPedigree: the origins of words from nature"Early Sanskritization. Origins and Development of the Kuru State"10.11588/ejvs.1995.4.823The Sanskrit epics, Part 2The City in South AsiaThe UpanishadsAn Introduction to HinduismReligions of the World, Second Edition: A Comprehensive Encyclopedia of Beliefs and PracticesA History of Ancient and Early Medieval India: From the Stone Age to the 12th CenturyEarly India: From the Origins to AD 1300Republics in ancient Indiapp. 83ff"Magadha Empire""Lumbini Development Trust: Restoring the Lumbini Garden"the originalThe great armies of antiquityarchived"The Achaemenid Persian Empire (550–330 B.C.)""East–West Orientation of Historical Empires"1076-156X"Dinner on the Grand Trunk Road"10.2307/32502263250226"Silappathikaram Tamil Literature"the originalManimekalai – English transliteration of Tamil originalIndian Temple Architecture: Form and Transformation : the Karṇāṭa Drāviḍa Tradition, 7th to 13th CenturiesBuddhist ArchitectureA History of India"The World Economy (GDP) : Historical Statistics by Professor Angus Maddison"The World Economy – Volume 1: A Millennial Perspective and Volume 2: Historical Statistics10.2307/32502140004-36483250214A Comprehensive History of India: Volume 2Between the Empires: Society in India, 300 to 400Emergence of Viṣṇu and Śiva Images in India: Numismatic and Sculptural Evidence"Parthian Pair of Earrings"the originalThe Medical Times and Gazette, Volume 1Greatest emporium in the worldThe Cambridge History of Ancient China: From the Origins of Civilization to 221 BCBuddhist Records of the Western WorldArchaeology in Soviet Central AsiaThe Grandeur of Gandhara: The Ancient Buddhist Civilization of the Swat, Peshawar, Kabul and Indus ValleysIndian Sculpture: Circa 500 B.C.-A.D. 700"The History of Pakistan: The Kushans"Gupta Dynasty – MSN Encartathe original"India – Historical Setting – The Classical Age – Gupta and Harsha""Gupta Dynasty, Golden Age Of India"the original"The Age of the Guptas and After"the originalNumber Theory and Its History"Gupta dynasty (Indian dynasty)""Gupta dynasty: empire in 4th century"the original"The Story of India – Photo Gallery"The ASI say499315420"Pallava script"p. 145"CNG: eAuction 329. INDIA, Post-Gupta (Ganges Valley). Vardhanas of Thanesar and Kanauj. Harshavardhana. Circa AD 606–647. AR Drachm (13mm, 2.28 g, 1h)""Harsha""Sthanvishvara (historical region, India)""Harsha (Indian emperor)"Shyama Kumar Chattopadhyaya (2000) The Philosophy of Sankar's Advaita VedantaShankara's IntroductionShankara's Introduction19373677Shankara's IntroductionIs The Buddhist 'No-Self' Doctrine Compatible With Pursuing Nirvana?The Seven Spiritual Laws Of YogaIndia: The Ancient Past. A History of the Indian-Subcontinent from 7000 BC to AD 1200The Kashmir Series: Glimpses of Kashmiri Culture – Vivekananda Kendra, Kanyakumari (p. 57).Al-Hind: Early Medieval India and the Expansion of Islam, 7th–11th CenturiesHistory of GopāchalaLand of Two Rivers: A History of Bengal from the Mahabharata to MujibEuropean Trade and Colonial ConquestA History of IndiaA Comprehensive History Of Ancient India (3 Vol. Set)"The Last Years of Cholas: The decline and fall of a dynasty"the originalFascinating Hindutva: Saffron Politics and Dalit MobilisationGazetteer of the province of OudhAl- Hind: The slave kings and the Islamic conquest. 2"Shahi Family"The Cambridge history of Islam"Ameer Nasir-ood-deen Subooktugeen"Gazetteer of the Attock District, 1930, Part 1Land of seven rivers: History of India's GeographyTemple Desecration and Indo-Muslim StatesIslam in South Asia: A Short HistoryBeyond Orientalism: The Work of Wilhelm Halbfass and Its Impact on Indian and Cross-cultural StudiesThe Making of Terrorism in Pakistan: Historical and Social Roots of ExtremismOrnament in Indian ArchitectureA historical review of Hindu India: 300 B.C. to 1200 A.D."Indian States and Union Territories"Islam in South Asia: A Short HistoryA Brief History of the Indian PeoplesThe Modern ReviewDelhi Sultanate"Battuta's Travels: Delhi, capital of Muslim India"the original"Timur – conquest of India"the originalIndia HandbookBhaktiThe Four Denomination of Hinduism10.1007/s11407-008-9049-925691067"Vijayanagara Research Project::Elephant Stables"10.2307/26465262646526Historical Dictionary of the TamilsBihar General Knowledge DigestMapping Bihar: From Medieval to Modern TimesPopular Literature and Pre-modern Societies in South AsiaA manual of the Kistna district in the presidency of MadrasAncient Indian History and CivilizationFragmented Memories: Struggling to be Tai-Ahom in India"The Islamic World to 1600: Rise of the Great Islamic Empires (The Mughal Empire)"the originalDynasties: A Global History of Power, 1300–1800, p. 105"Whose fort is it anyway"10.1111/0020-8833.000532600793Development Centre Studies The World Economy Historical Statistics: Historical Statistics"India's Deindustrialization in the 18th and 19th Centuries"The Mughal Empire, p. 190"The Long Globalization and Textile Producers in India"The Mughal World: Life in India's Last Golden AgeAurangzeb: The Life and Legacy of India's Most Controversial KingIn the Shadow of the Taj: A Portrait of Agra"Iran in the Age of the Raj"p. 8610.2307/20539802053980Delhi, the Capital of IndiaAn Advanced History of Modern India"Journal of the Tanjore Maharaja Serfoji's Sarasvati Mahal Library"The Rediscovery of India: A New SubcontinentIslamic Renaissance In South Asia (1707–1867) : The Role Of Shah Waliallah & His SuccessorsAn Advanced History of Modern IndiaThe Great Maratha Mahadaji Scindia"Full text of "Selections from the papers of Lord Metcalfe; late governor-general of India, governor of Jamaica, and governor-general of Canada""The Discovery Of IndiaThe Sacred City of the Hindus: An Account of Benares in Ancient and Modern TimesResurrecting Banaras: Urban Space, Architecture and Religious BoundariesFaith & Philosophy of Sikhism"Missiles mainstay of Pak's N-arsenal"History Modern India By S.N. Sen"Sirajuddaula"ArchivedLongman History & Civics (Dual Government in Bengal)Madhya Pradesh National Means-Cum-Merit Scholarship Exam (Warren Hasting's system of Dual Government)A Military History of Britain: from 1775 to the PresentIndian Cultural Heritage Perspective For TourismHindu Rulers, Muslim Subjects: Islam, Rights, and the History of KashmirIndian HistoryAn Atlas and Survey of South Asian HistoryAn Historical Account of the British Trade Over the Caspian SeaIndian Merchants and Eurasian Trade, 1600–1750The Indian diaspora in Central Asia and its trade, 1550–1900From Constantinople to the home of Omar Khayyam: travels in Transcaucasia and northern Persia for historic and literary researchA journey from Bengal to England: through the northern part of India, Kashmire, Afghanistan, and Persia, and into Russia, by the Caspian-SeaA Second Journey through Persia, Armenia, and Asia Minor, to Constantinople, between the Years 1810 and 1816Reports from the consuls of the United States, 1887Portugal and its Empire, 1250–1800 (Collected Essays in Memory of Glenn J. Ames).: Portuguese Studies Review, Vol. 17, No. 1The Dutch Power in Kerala, 1729–1758http://mod.nic.inArchivedDossier Goa – A Recusa do Sacrifício InútilAn Imperial Crisis in British India: The Manipur Uprising of 1891The Truth of Babri Mosque"Kolkata (Calcutta) : History"the original"Robert Clive, Baron Clive, 'Clive of India', 1725–1774""The Transformation from a Pre-Colonial to a Colonial Order: The Case of India"10.2307/25955872595587A versatile geniusArchived10.1109/MWSYM.1997.602854"Rabindranath Tagore on Education"the original"Essay on 'Derozio and the Young Bengal Movement'"Poverty and Famines: An Essay on Entitlement and Deprivation"Plague"the originalPopulation Growth and Land Use"Reintegrating India with the World Economy""Census Of India 1931"A history of modern India, 1480–1950"'India's well-timed diversification of army helped democracy' | Business Standard News"Bal Gangadhar Tilak: Struggle for Swaraj"Participants from the Indian subcontinent in the First World War""Commonwealth War Graves Commission Annual Report 2007–2008 Online"the original1462689197110.1017/s0010417500016534178920Eurocentrism: a marxian critical realist critique"Ranjit Guha, "On Some Aspects of Historiography of Colonial India""10.2307/2168385216838510.7202/016593ar"Harvard scholar says the idea of India dates to a much earlier time than the British or the Mughals""In The Footsteps of Pilgrims""India's spiritual landscape: The heavens and the earth""India: A Sacred Geography by Diana L Eck – review"Modern India: The Origins of an Asian Democracy10.2307/21694222169422964322464"The Indian Subcontinent and 'Out of Africa 1'"254043308Encyclopedia of World ReligionsThe Evolution and History of Human Populations in South Asia: Inter-disciplinary Studies in Archaeology, Biological Anthropology, Linguistics and Genetics"The Early Paleolithic of the Indian Subcontinent: Hominin Colonization, Dispersals and Occupation History"Ancient Indian History and CivilizationAncient Indian Social History: Some Interpretationsthe originalIndia Before EuropeA Concise History of Modern India"The beginning of the historical period, c. 500–150 BCE"full textA History of Indiathe originalexcerpt and text searchexcerptexcerptAn Economic History of India: From Pre-Colonial Times to 1991excerpt and text searchexcerpt and text searchIndia as known to the ancient world10.1111/j.1468-0289.1985.tb00391.x2597191onlineThe History of India, as told by its own historians. The Muhammadan Periodonline editionHans William Brown research collection on 19th-century missionary work in India, 1882–1932, Ms. Coll. 1033, Kislak Center for Special Collections, Rare Books and Manuscripts, University of Pennsylvaniaee

                                Image
                                AfghanistanArmeniaAzerbaijanBahrainBangladeshBhutanBruneiCambodiaChinaCyprusEast Timor (Timor-Leste)EgyptGeorgiaIndiaIndonesiaIranIraqIsraelJapanJordanKazakhstanNorth KoreaSouth KoreaKuwaitKyrgyzstanLaosLebanonMalaysiaMaldivesMongoliaMyanmarNepalOmanPakistanPhilippinesQatarRussiaSaudi ArabiaSingaporeSri LankaSyriaTajikistanThailandTurkeyTurkmenistanUnited Arab EmiratesUzbekistanVietnamYemenBritish Indian Ocean TerritoryChristmas IslandCocos (Keeling) IslandsHong KongMacau History of IndiaHistory of South Asia Anatomically modern humansfarmingpastoralismwheatbarleygoats, sheep, and cattleIndus Valley Civilizationcradle of civilisationMature Harappanhandicraftcarnelian productsseal carvingmetallurgyurban planningbaked brick housesdrainage systemswater supply systemsIron AgeVedic Civilisationpersistent droughtIndo-Aryan tribeswaves of Aryan migrationVedic periodVedasHinduismJanapadas (monarchical, state-level polities)casteBrahminsKshatriyasVaishyasShudrasIndo-Gangetic plainMahajanapadas ...
                                Read more

                                map list to bin numbersquantilization (if that is a word)Map efficiently over duplicates in listGenerating a list of cubefree numbersIs there an equivalent to MATLAB's linspace?Convert a list of hexadecimal numbers to decimalTaking one list Mod a second listlist of items and group of alternative itemsHow find numbers in this list of inequalities?Selecting list entries with a True False index list of similar lengthReplace element in array by checking condition in another listAttempting to fill a table with the number of elements in each bin and make a table with the elements in the bins?

                                Image
                                Does the average primeness of natural numbers tend to zero? New order #4: World How can I fix this gap between bookcases I made? How could a lack of term limits lead to a "dictatorship?" Why do UK politicians seemingly ignore opinion polls on Brexit? COUNT(*) or MAX(id) - which is faster? How can I add custom success page Extreme, but not acceptable situation and I can't start the work tomorrow morning Does bootstrapped regression allow for inference? Is it wise to focus on putting odd beats on left when playing double bass drums? What happens when a metallic dragon and a chromatic dragon mate? Why is the design of haulage companies so “special”? "listening to me about as much as you're listening to this pole here" What does "enim et" mean? I’m planning on buying a laser printer but concerned about the life cycle of toner in the machine Domain expired, GoDaddy holds it and is asking more money Does it makes sense to ...
                                Read more