Limit involving integrals [on hold]











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I am completely clueless about the approach to solve this question. I tried applying l ho'pital rule but am getting stuck.



Kindly help.






limits







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put on hold as off-topic by José Carlos Santos, amWhy, RRL, TheGeekGreek, Chris Godsil 2 days ago


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    up vote
    -2
    down vote

    favorite












    enter image description here



    I am completely clueless about the approach to solve this question. I tried applying l ho'pital rule but am getting stuck.



    Kindly help.






    limits







    share|cite|improve this question













    put on hold as off-topic by José Carlos Santos, amWhy, RRL, TheGeekGreek, Chris Godsil 2 days ago


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, amWhy, RRL, TheGeekGreek, Chris Godsil

    If this question can be reworded to fit the rules in the help center, please edit the question.















      up vote
      -2
      down vote

      favorite









      up vote
      -2
      down vote

      favorite











      enter image description here



      I am completely clueless about the approach to solve this question. I tried applying l ho'pital rule but am getting stuck.



      Kindly help.






      limits







      share|cite|improve this question













      enter image description here



      I am completely clueless about the approach to solve this question. I tried applying l ho'pital rule but am getting stuck.



      Kindly help.






      limits




      limits






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 2 days ago









      Akash Gautama

      915




      915




      put on hold as off-topic by José Carlos Santos, amWhy, RRL, TheGeekGreek, Chris Godsil 2 days ago


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, amWhy, RRL, TheGeekGreek, Chris Godsil

      If this question can be reworded to fit the rules in the help center, please edit the question.




      put on hold as off-topic by José Carlos Santos, amWhy, RRL, TheGeekGreek, Chris Godsil 2 days ago


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, amWhy, RRL, TheGeekGreek, Chris Godsil

      If this question can be reworded to fit the rules in the help center, please edit the question.






















          1 Answer
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          HINT:



          Write $frac1{(1+x^2)^n} =e^{-nlog(1+x^2)}$.



          Then note that all of the contribution to the value of the limit of interest comes from the integration around $0$.



          So, heuristically we can approximate the integral of interest with its lower bound



          $$sqrt n int_0^1 e^{-nlog(1+x)^2},dx approx sqrt n int_0^1 e^{-n x^2},dx$$



          Can you finish now?




          Alternatively, to be rigorous, just let $x=y/sqrt n$ in the original integral and apply the Dominated Convergence Theorem.






          share|cite|improve this answer






























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote













            HINT:



            Write $frac1{(1+x^2)^n} =e^{-nlog(1+x^2)}$.



            Then note that all of the contribution to the value of the limit of interest comes from the integration around $0$.



            So, heuristically we can approximate the integral of interest with its lower bound



            $$sqrt n int_0^1 e^{-nlog(1+x)^2},dx approx sqrt n int_0^1 e^{-n x^2},dx$$



            Can you finish now?




            Alternatively, to be rigorous, just let $x=y/sqrt n$ in the original integral and apply the Dominated Convergence Theorem.






            share|cite|improve this answer



























              up vote
              1
              down vote













              HINT:



              Write $frac1{(1+x^2)^n} =e^{-nlog(1+x^2)}$.



              Then note that all of the contribution to the value of the limit of interest comes from the integration around $0$.



              So, heuristically we can approximate the integral of interest with its lower bound



              $$sqrt n int_0^1 e^{-nlog(1+x)^2},dx approx sqrt n int_0^1 e^{-n x^2},dx$$



              Can you finish now?




              Alternatively, to be rigorous, just let $x=y/sqrt n$ in the original integral and apply the Dominated Convergence Theorem.






              share|cite|improve this answer

























                up vote
                1
                down vote










                up vote
                1
                down vote









                HINT:



                Write $frac1{(1+x^2)^n} =e^{-nlog(1+x^2)}$.



                Then note that all of the contribution to the value of the limit of interest comes from the integration around $0$.



                So, heuristically we can approximate the integral of interest with its lower bound



                $$sqrt n int_0^1 e^{-nlog(1+x)^2},dx approx sqrt n int_0^1 e^{-n x^2},dx$$



                Can you finish now?




                Alternatively, to be rigorous, just let $x=y/sqrt n$ in the original integral and apply the Dominated Convergence Theorem.






                share|cite|improve this answer














                HINT:



                Write $frac1{(1+x^2)^n} =e^{-nlog(1+x^2)}$.



                Then note that all of the contribution to the value of the limit of interest comes from the integration around $0$.



                So, heuristically we can approximate the integral of interest with its lower bound



                $$sqrt n int_0^1 e^{-nlog(1+x)^2},dx approx sqrt n int_0^1 e^{-n x^2},dx$$



                Can you finish now?




                Alternatively, to be rigorous, just let $x=y/sqrt n$ in the original integral and apply the Dominated Convergence Theorem.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 2 days ago

























                answered 2 days ago









                Mark Viola

                129k1273170




                129k1273170















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