Finding $limlimits_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)$











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What is$$lim_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)?$$So it is$$lim_{n→∞}frac{n^3(sqrt{n^2+sqrt{n^4+1}})^2-(nsqrt{2})^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}.$$
I do not know what to do next, because my resuts is $∞$ but the answer from book is $dfrac{1}{4sqrt{2}}$.






calculus limits







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  • try putting n=1/x.then apply binomial expansions.
    – maveric
    Dec 4 at 10:07












  • @maveric That's of course an effective method but we don't really need that on this case.
    – gimusi
    Dec 4 at 10:08










  • How can be your result $infty$ from here $$lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$ Note that at the numerator we have an indeterminate form $infty-infty$ therefore you can't conclude form here.
    – gimusi
    Dec 4 at 10:14












  • @gimusi. Using the binomial expansion is a good way if you want to know how is approached the limit (this is just building the Taylor expansion). I hope and wish that we could have a discussion about this kind of problems one of these days. Cheers.
    – Claude Leibovici
    Dec 4 at 10:37












  • @ClaudeLeibovici Yes of course, I also like Taylor's very much but in that case I think that it suffices operate by agebraic methods also I think thet the OP is looking for such kind of solution. Anyway the suggestion is good as an alternative approach! Cheers
    – gimusi
    Dec 4 at 10:39















up vote
3
down vote

favorite












What is$$lim_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)?$$So it is$$lim_{n→∞}frac{n^3(sqrt{n^2+sqrt{n^4+1}})^2-(nsqrt{2})^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}.$$
I do not know what to do next, because my resuts is $∞$ but the answer from book is $dfrac{1}{4sqrt{2}}$.






calculus limits







share|cite|improve this question
























  • try putting n=1/x.then apply binomial expansions.
    – maveric
    Dec 4 at 10:07












  • @maveric That's of course an effective method but we don't really need that on this case.
    – gimusi
    Dec 4 at 10:08










  • How can be your result $infty$ from here $$lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$ Note that at the numerator we have an indeterminate form $infty-infty$ therefore you can't conclude form here.
    – gimusi
    Dec 4 at 10:14












  • @gimusi. Using the binomial expansion is a good way if you want to know how is approached the limit (this is just building the Taylor expansion). I hope and wish that we could have a discussion about this kind of problems one of these days. Cheers.
    – Claude Leibovici
    Dec 4 at 10:37












  • @ClaudeLeibovici Yes of course, I also like Taylor's very much but in that case I think that it suffices operate by agebraic methods also I think thet the OP is looking for such kind of solution. Anyway the suggestion is good as an alternative approach! Cheers
    – gimusi
    Dec 4 at 10:39













up vote
3
down vote

favorite









up vote
3
down vote

favorite











What is$$lim_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)?$$So it is$$lim_{n→∞}frac{n^3(sqrt{n^2+sqrt{n^4+1}})^2-(nsqrt{2})^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}.$$
I do not know what to do next, because my resuts is $∞$ but the answer from book is $dfrac{1}{4sqrt{2}}$.






calculus limits







share|cite|improve this question















What is$$lim_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)?$$So it is$$lim_{n→∞}frac{n^3(sqrt{n^2+sqrt{n^4+1}})^2-(nsqrt{2})^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}.$$
I do not know what to do next, because my resuts is $∞$ but the answer from book is $dfrac{1}{4sqrt{2}}$.






calculus limits




calculus limits






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share|cite|improve this question













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edited Dec 4 at 10:18









Jyrki Lahtonen

107k12166364




107k12166364










asked Dec 4 at 9:59









Estera Gmiterek

306




306












  • try putting n=1/x.then apply binomial expansions.
    – maveric
    Dec 4 at 10:07












  • @maveric That's of course an effective method but we don't really need that on this case.
    – gimusi
    Dec 4 at 10:08










  • How can be your result $infty$ from here $$lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$ Note that at the numerator we have an indeterminate form $infty-infty$ therefore you can't conclude form here.
    – gimusi
    Dec 4 at 10:14












  • @gimusi. Using the binomial expansion is a good way if you want to know how is approached the limit (this is just building the Taylor expansion). I hope and wish that we could have a discussion about this kind of problems one of these days. Cheers.
    – Claude Leibovici
    Dec 4 at 10:37












  • @ClaudeLeibovici Yes of course, I also like Taylor's very much but in that case I think that it suffices operate by agebraic methods also I think thet the OP is looking for such kind of solution. Anyway the suggestion is good as an alternative approach! Cheers
    – gimusi
    Dec 4 at 10:39


















  • try putting n=1/x.then apply binomial expansions.
    – maveric
    Dec 4 at 10:07












  • @maveric That's of course an effective method but we don't really need that on this case.
    – gimusi
    Dec 4 at 10:08










  • How can be your result $infty$ from here $$lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$ Note that at the numerator we have an indeterminate form $infty-infty$ therefore you can't conclude form here.
    – gimusi
    Dec 4 at 10:14












  • @gimusi. Using the binomial expansion is a good way if you want to know how is approached the limit (this is just building the Taylor expansion). I hope and wish that we could have a discussion about this kind of problems one of these days. Cheers.
    – Claude Leibovici
    Dec 4 at 10:37












  • @ClaudeLeibovici Yes of course, I also like Taylor's very much but in that case I think that it suffices operate by agebraic methods also I think thet the OP is looking for such kind of solution. Anyway the suggestion is good as an alternative approach! Cheers
    – gimusi
    Dec 4 at 10:39
















try putting n=1/x.then apply binomial expansions.
– maveric
Dec 4 at 10:07






try putting n=1/x.then apply binomial expansions.
– maveric
Dec 4 at 10:07














@maveric That's of course an effective method but we don't really need that on this case.
– gimusi
Dec 4 at 10:08




@maveric That's of course an effective method but we don't really need that on this case.
– gimusi
Dec 4 at 10:08












How can be your result $infty$ from here $$lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$ Note that at the numerator we have an indeterminate form $infty-infty$ therefore you can't conclude form here.
– gimusi
Dec 4 at 10:14






How can be your result $infty$ from here $$lim_{n→∞}frac{n^3(n^2+sqrt{n^4+1}-2n^2)}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$ Note that at the numerator we have an indeterminate form $infty-infty$ therefore you can't conclude form here.
– gimusi
Dec 4 at 10:14














@gimusi. Using the binomial expansion is a good way if you want to know how is approached the limit (this is just building the Taylor expansion). I hope and wish that we could have a discussion about this kind of problems one of these days. Cheers.
– Claude Leibovici
Dec 4 at 10:37






@gimusi. Using the binomial expansion is a good way if you want to know how is approached the limit (this is just building the Taylor expansion). I hope and wish that we could have a discussion about this kind of problems one of these days. Cheers.
– Claude Leibovici
Dec 4 at 10:37














@ClaudeLeibovici Yes of course, I also like Taylor's very much but in that case I think that it suffices operate by agebraic methods also I think thet the OP is looking for such kind of solution. Anyway the suggestion is good as an alternative approach! Cheers
– gimusi
Dec 4 at 10:39




@ClaudeLeibovici Yes of course, I also like Taylor's very much but in that case I think that it suffices operate by agebraic methods also I think thet the OP is looking for such kind of solution. Anyway the suggestion is good as an alternative approach! Cheers
– gimusi
Dec 4 at 10:39










5 Answers
5






active

oldest

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up vote
4
down vote



accepted










HINT



You only need to apply the trick twice, indeed we have that



$$sqrt{n^2+sqrt{n^4+1}}-nsqrt{2}=(sqrt{n^2+sqrt{n^4+1}}-nsqrt{2})cdotfrac{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=$$$$=frac{n^2+sqrt{n^4+1}-2n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$



and



$$frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}cdot frac{sqrt{n^4+1}+n^2}{sqrt{n^4+1}+n^2}=$$$$=frac{1}{(sqrt{n^2+sqrt{n^4+1}}+nsqrt{2})(sqrt{n^4+1}+n^2)}$$



Can you conclude form here?






share|cite|improve this answer























  • Yes, thanks i understand !
    – Estera Gmiterek
    Dec 4 at 19:13










  • That’s nice! Well done, you are welcome! Bye
    – gimusi
    Dec 4 at 19:17


















up vote
2
down vote













Let $1/n=h$



$$lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



$$=lim_{hto0^+}dfrac{1+sqrt{1+h^4}-2}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



$$=lim_{hto0^+}dfrac{1+h^4-1}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+h^4}+1}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



$$=dfrac1{(sqrt1+1)(sqrt{1+sqrt1}+sqrt2)}$$






share|cite|improve this answer























  • Nice way to simplify the manipulation. (+1)
    – gimusi
    Dec 4 at 10:12










  • Sorry, but the final result is wrong
    – egreg
    Dec 4 at 10:28










  • @egreg, Thanks for your feedback. Rectfied.
    – lab bhattacharjee
    Dec 4 at 10:33










  • Related : math.stackexchange.com/questions/524288/…
    – lab bhattacharjee
    Dec 7 at 5:10


















up vote
1
down vote













The expedite way:



$$sqrt{1+sqrt{1+n^{-4}}}=sqrt{1+1+dfrac12n^{-4}+o(n^{-4})}=sqrt2sqrt{1+dfrac14n^{-4}+o(n^{-4})}=sqrt2left(1+dfrac18n^{-4}+o(n^{-4})right)$$



and the limit is



$$frac{sqrt2}8.$$






share|cite|improve this answer




























    up vote
    0
    down vote













    First replace $1/x=h$ to find



    $$L=lim_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)=lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



    Let $sqrt{1+sqrt{1+h^4}}=y+sqrt2implies1+sqrt{1+h^4}=(sqrt2+y)^2=2+2sqrt2y+y^2$



    $$1+h^4=(1+y(2sqrt2+y))^2=1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2$$



    $$L=lim_{yto0}dfrac{1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2-1}y=?$$






    share|cite|improve this answer




























      up vote
      0
      down vote













      Formally substitute $n=1/t$; if the function you get has a limit for $tto0^+$, then it is the same as the limit you are looking for. So consider
      $$
      lim_{tto0^+}frac{1}{t^3}left(
      sqrt{frac{1}{t^2}+sqrt{frac{1}{t^4}+1}}-frac{sqrt{2}}{t}
      right)=
      lim_{tto0^+}frac{sqrt{1+sqrt{1+t^4}}-sqrt{2}}{t^4}
      $$
      Now the dependency is only on $t^4$, so the limit is the same as
      $$
      lim_{uto0^+}frac{sqrt{1+sqrt{1+u}}-sqrt{2}}{u}
      $$
      which is the derivative at $0$ of $f(u)=sqrt{1+sqrt{1+u}}$.




      Since $$f'(u)=frac{1}{2sqrt{1+sqrt{1+u}}}frac{1}{2sqrt{1+u}}$$ we have $$f'(0)=frac{1}{4sqrt{2}}$$







      share|cite|improve this answer























      • As always I don't agree that you give a full answer. In my opinion it is not useful for the asker nor for the site. I don't downvote because it is of course a correct and good answer. But virtually I give you a "-1".
        – gimusi
        Dec 4 at 10:31













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      5 Answers
      5






      active

      oldest

      votes








      5 Answers
      5






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      4
      down vote



      accepted










      HINT



      You only need to apply the trick twice, indeed we have that



      $$sqrt{n^2+sqrt{n^4+1}}-nsqrt{2}=(sqrt{n^2+sqrt{n^4+1}}-nsqrt{2})cdotfrac{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=$$$$=frac{n^2+sqrt{n^4+1}-2n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$



      and



      $$frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}cdot frac{sqrt{n^4+1}+n^2}{sqrt{n^4+1}+n^2}=$$$$=frac{1}{(sqrt{n^2+sqrt{n^4+1}}+nsqrt{2})(sqrt{n^4+1}+n^2)}$$



      Can you conclude form here?






      share|cite|improve this answer























      • Yes, thanks i understand !
        – Estera Gmiterek
        Dec 4 at 19:13










      • That’s nice! Well done, you are welcome! Bye
        – gimusi
        Dec 4 at 19:17















      up vote
      4
      down vote



      accepted










      HINT



      You only need to apply the trick twice, indeed we have that



      $$sqrt{n^2+sqrt{n^4+1}}-nsqrt{2}=(sqrt{n^2+sqrt{n^4+1}}-nsqrt{2})cdotfrac{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=$$$$=frac{n^2+sqrt{n^4+1}-2n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$



      and



      $$frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}cdot frac{sqrt{n^4+1}+n^2}{sqrt{n^4+1}+n^2}=$$$$=frac{1}{(sqrt{n^2+sqrt{n^4+1}}+nsqrt{2})(sqrt{n^4+1}+n^2)}$$



      Can you conclude form here?






      share|cite|improve this answer























      • Yes, thanks i understand !
        – Estera Gmiterek
        Dec 4 at 19:13










      • That’s nice! Well done, you are welcome! Bye
        – gimusi
        Dec 4 at 19:17













      up vote
      4
      down vote



      accepted







      up vote
      4
      down vote



      accepted






      HINT



      You only need to apply the trick twice, indeed we have that



      $$sqrt{n^2+sqrt{n^4+1}}-nsqrt{2}=(sqrt{n^2+sqrt{n^4+1}}-nsqrt{2})cdotfrac{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=$$$$=frac{n^2+sqrt{n^4+1}-2n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$



      and



      $$frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}cdot frac{sqrt{n^4+1}+n^2}{sqrt{n^4+1}+n^2}=$$$$=frac{1}{(sqrt{n^2+sqrt{n^4+1}}+nsqrt{2})(sqrt{n^4+1}+n^2)}$$



      Can you conclude form here?






      share|cite|improve this answer














      HINT



      You only need to apply the trick twice, indeed we have that



      $$sqrt{n^2+sqrt{n^4+1}}-nsqrt{2}=(sqrt{n^2+sqrt{n^4+1}}-nsqrt{2})cdotfrac{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=$$$$=frac{n^2+sqrt{n^4+1}-2n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}$$



      and



      $$frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}=frac{sqrt{n^4+1}-n^2}{sqrt{n^2+sqrt{n^4+1}}+nsqrt{2}}cdot frac{sqrt{n^4+1}+n^2}{sqrt{n^4+1}+n^2}=$$$$=frac{1}{(sqrt{n^2+sqrt{n^4+1}}+nsqrt{2})(sqrt{n^4+1}+n^2)}$$



      Can you conclude form here?







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Dec 4 at 10:10

























      answered Dec 4 at 10:05









      gimusi

      91.9k84495




      91.9k84495












      • Yes, thanks i understand !
        – Estera Gmiterek
        Dec 4 at 19:13










      • That’s nice! Well done, you are welcome! Bye
        – gimusi
        Dec 4 at 19:17


















      • Yes, thanks i understand !
        – Estera Gmiterek
        Dec 4 at 19:13










      • That’s nice! Well done, you are welcome! Bye
        – gimusi
        Dec 4 at 19:17
















      Yes, thanks i understand !
      – Estera Gmiterek
      Dec 4 at 19:13




      Yes, thanks i understand !
      – Estera Gmiterek
      Dec 4 at 19:13












      That’s nice! Well done, you are welcome! Bye
      – gimusi
      Dec 4 at 19:17




      That’s nice! Well done, you are welcome! Bye
      – gimusi
      Dec 4 at 19:17










      up vote
      2
      down vote













      Let $1/n=h$



      $$lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



      $$=lim_{hto0^+}dfrac{1+sqrt{1+h^4}-2}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



      $$=lim_{hto0^+}dfrac{1+h^4-1}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+h^4}+1}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



      $$=dfrac1{(sqrt1+1)(sqrt{1+sqrt1}+sqrt2)}$$






      share|cite|improve this answer























      • Nice way to simplify the manipulation. (+1)
        – gimusi
        Dec 4 at 10:12










      • Sorry, but the final result is wrong
        – egreg
        Dec 4 at 10:28










      • @egreg, Thanks for your feedback. Rectfied.
        – lab bhattacharjee
        Dec 4 at 10:33










      • Related : math.stackexchange.com/questions/524288/…
        – lab bhattacharjee
        Dec 7 at 5:10















      up vote
      2
      down vote













      Let $1/n=h$



      $$lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



      $$=lim_{hto0^+}dfrac{1+sqrt{1+h^4}-2}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



      $$=lim_{hto0^+}dfrac{1+h^4-1}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+h^4}+1}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



      $$=dfrac1{(sqrt1+1)(sqrt{1+sqrt1}+sqrt2)}$$






      share|cite|improve this answer























      • Nice way to simplify the manipulation. (+1)
        – gimusi
        Dec 4 at 10:12










      • Sorry, but the final result is wrong
        – egreg
        Dec 4 at 10:28










      • @egreg, Thanks for your feedback. Rectfied.
        – lab bhattacharjee
        Dec 4 at 10:33










      • Related : math.stackexchange.com/questions/524288/…
        – lab bhattacharjee
        Dec 7 at 5:10













      up vote
      2
      down vote










      up vote
      2
      down vote









      Let $1/n=h$



      $$lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



      $$=lim_{hto0^+}dfrac{1+sqrt{1+h^4}-2}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



      $$=lim_{hto0^+}dfrac{1+h^4-1}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+h^4}+1}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



      $$=dfrac1{(sqrt1+1)(sqrt{1+sqrt1}+sqrt2)}$$






      share|cite|improve this answer














      Let $1/n=h$



      $$lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



      $$=lim_{hto0^+}dfrac{1+sqrt{1+h^4}-2}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



      $$=lim_{hto0^+}dfrac{1+h^4-1}{h^4}cdotlim_{hto0^+}dfrac1{sqrt{1+h^4}+1}cdotlim_{hto0^+}dfrac1{sqrt{1+sqrt{1+h^4}}+sqrt2}$$



      $$=dfrac1{(sqrt1+1)(sqrt{1+sqrt1}+sqrt2)}$$







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Dec 4 at 10:33

























      answered Dec 4 at 10:08









      lab bhattacharjee

      222k15155273




      222k15155273












      • Nice way to simplify the manipulation. (+1)
        – gimusi
        Dec 4 at 10:12










      • Sorry, but the final result is wrong
        – egreg
        Dec 4 at 10:28










      • @egreg, Thanks for your feedback. Rectfied.
        – lab bhattacharjee
        Dec 4 at 10:33










      • Related : math.stackexchange.com/questions/524288/…
        – lab bhattacharjee
        Dec 7 at 5:10


















      • Nice way to simplify the manipulation. (+1)
        – gimusi
        Dec 4 at 10:12










      • Sorry, but the final result is wrong
        – egreg
        Dec 4 at 10:28










      • @egreg, Thanks for your feedback. Rectfied.
        – lab bhattacharjee
        Dec 4 at 10:33










      • Related : math.stackexchange.com/questions/524288/…
        – lab bhattacharjee
        Dec 7 at 5:10
















      Nice way to simplify the manipulation. (+1)
      – gimusi
      Dec 4 at 10:12




      Nice way to simplify the manipulation. (+1)
      – gimusi
      Dec 4 at 10:12












      Sorry, but the final result is wrong
      – egreg
      Dec 4 at 10:28




      Sorry, but the final result is wrong
      – egreg
      Dec 4 at 10:28












      @egreg, Thanks for your feedback. Rectfied.
      – lab bhattacharjee
      Dec 4 at 10:33




      @egreg, Thanks for your feedback. Rectfied.
      – lab bhattacharjee
      Dec 4 at 10:33












      Related : math.stackexchange.com/questions/524288/…
      – lab bhattacharjee
      Dec 7 at 5:10




      Related : math.stackexchange.com/questions/524288/…
      – lab bhattacharjee
      Dec 7 at 5:10










      up vote
      1
      down vote













      The expedite way:



      $$sqrt{1+sqrt{1+n^{-4}}}=sqrt{1+1+dfrac12n^{-4}+o(n^{-4})}=sqrt2sqrt{1+dfrac14n^{-4}+o(n^{-4})}=sqrt2left(1+dfrac18n^{-4}+o(n^{-4})right)$$



      and the limit is



      $$frac{sqrt2}8.$$






      share|cite|improve this answer

























        up vote
        1
        down vote













        The expedite way:



        $$sqrt{1+sqrt{1+n^{-4}}}=sqrt{1+1+dfrac12n^{-4}+o(n^{-4})}=sqrt2sqrt{1+dfrac14n^{-4}+o(n^{-4})}=sqrt2left(1+dfrac18n^{-4}+o(n^{-4})right)$$



        and the limit is



        $$frac{sqrt2}8.$$






        share|cite|improve this answer























          up vote
          1
          down vote










          up vote
          1
          down vote









          The expedite way:



          $$sqrt{1+sqrt{1+n^{-4}}}=sqrt{1+1+dfrac12n^{-4}+o(n^{-4})}=sqrt2sqrt{1+dfrac14n^{-4}+o(n^{-4})}=sqrt2left(1+dfrac18n^{-4}+o(n^{-4})right)$$



          and the limit is



          $$frac{sqrt2}8.$$






          share|cite|improve this answer












          The expedite way:



          $$sqrt{1+sqrt{1+n^{-4}}}=sqrt{1+1+dfrac12n^{-4}+o(n^{-4})}=sqrt2sqrt{1+dfrac14n^{-4}+o(n^{-4})}=sqrt2left(1+dfrac18n^{-4}+o(n^{-4})right)$$



          and the limit is



          $$frac{sqrt2}8.$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 4 at 10:31









          Yves Daoust

          123k668219




          123k668219






















              up vote
              0
              down vote













              First replace $1/x=h$ to find



              $$L=lim_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)=lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



              Let $sqrt{1+sqrt{1+h^4}}=y+sqrt2implies1+sqrt{1+h^4}=(sqrt2+y)^2=2+2sqrt2y+y^2$



              $$1+h^4=(1+y(2sqrt2+y))^2=1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2$$



              $$L=lim_{yto0}dfrac{1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2-1}y=?$$






              share|cite|improve this answer

























                up vote
                0
                down vote













                First replace $1/x=h$ to find



                $$L=lim_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)=lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



                Let $sqrt{1+sqrt{1+h^4}}=y+sqrt2implies1+sqrt{1+h^4}=(sqrt2+y)^2=2+2sqrt2y+y^2$



                $$1+h^4=(1+y(2sqrt2+y))^2=1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2$$



                $$L=lim_{yto0}dfrac{1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2-1}y=?$$






                share|cite|improve this answer























                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  First replace $1/x=h$ to find



                  $$L=lim_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)=lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



                  Let $sqrt{1+sqrt{1+h^4}}=y+sqrt2implies1+sqrt{1+h^4}=(sqrt2+y)^2=2+2sqrt2y+y^2$



                  $$1+h^4=(1+y(2sqrt2+y))^2=1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2$$



                  $$L=lim_{yto0}dfrac{1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2-1}y=?$$






                  share|cite|improve this answer












                  First replace $1/x=h$ to find



                  $$L=lim_{n→∞}n^3(sqrt{n^2+sqrt{n^4+1}}-nsqrt2)=lim_{hto0^+}dfrac{sqrt{1+sqrt{1+h^4}}-sqrt2}{h^4}$$



                  Let $sqrt{1+sqrt{1+h^4}}=y+sqrt2implies1+sqrt{1+h^4}=(sqrt2+y)^2=2+2sqrt2y+y^2$



                  $$1+h^4=(1+y(2sqrt2+y))^2=1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2$$



                  $$L=lim_{yto0}dfrac{1+2y(2sqrt2+y)+y^2(2sqrt2+y)^2-1}y=?$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 4 at 10:16









                  lab bhattacharjee

                  222k15155273




                  222k15155273






















                      up vote
                      0
                      down vote













                      Formally substitute $n=1/t$; if the function you get has a limit for $tto0^+$, then it is the same as the limit you are looking for. So consider
                      $$
                      lim_{tto0^+}frac{1}{t^3}left(
                      sqrt{frac{1}{t^2}+sqrt{frac{1}{t^4}+1}}-frac{sqrt{2}}{t}
                      right)=
                      lim_{tto0^+}frac{sqrt{1+sqrt{1+t^4}}-sqrt{2}}{t^4}
                      $$
                      Now the dependency is only on $t^4$, so the limit is the same as
                      $$
                      lim_{uto0^+}frac{sqrt{1+sqrt{1+u}}-sqrt{2}}{u}
                      $$
                      which is the derivative at $0$ of $f(u)=sqrt{1+sqrt{1+u}}$.




                      Since $$f'(u)=frac{1}{2sqrt{1+sqrt{1+u}}}frac{1}{2sqrt{1+u}}$$ we have $$f'(0)=frac{1}{4sqrt{2}}$$







                      share|cite|improve this answer























                      • As always I don't agree that you give a full answer. In my opinion it is not useful for the asker nor for the site. I don't downvote because it is of course a correct and good answer. But virtually I give you a "-1".
                        – gimusi
                        Dec 4 at 10:31

















                      up vote
                      0
                      down vote













                      Formally substitute $n=1/t$; if the function you get has a limit for $tto0^+$, then it is the same as the limit you are looking for. So consider
                      $$
                      lim_{tto0^+}frac{1}{t^3}left(
                      sqrt{frac{1}{t^2}+sqrt{frac{1}{t^4}+1}}-frac{sqrt{2}}{t}
                      right)=
                      lim_{tto0^+}frac{sqrt{1+sqrt{1+t^4}}-sqrt{2}}{t^4}
                      $$
                      Now the dependency is only on $t^4$, so the limit is the same as
                      $$
                      lim_{uto0^+}frac{sqrt{1+sqrt{1+u}}-sqrt{2}}{u}
                      $$
                      which is the derivative at $0$ of $f(u)=sqrt{1+sqrt{1+u}}$.




                      Since $$f'(u)=frac{1}{2sqrt{1+sqrt{1+u}}}frac{1}{2sqrt{1+u}}$$ we have $$f'(0)=frac{1}{4sqrt{2}}$$







                      share|cite|improve this answer























                      • As always I don't agree that you give a full answer. In my opinion it is not useful for the asker nor for the site. I don't downvote because it is of course a correct and good answer. But virtually I give you a "-1".
                        – gimusi
                        Dec 4 at 10:31















                      up vote
                      0
                      down vote










                      up vote
                      0
                      down vote









                      Formally substitute $n=1/t$; if the function you get has a limit for $tto0^+$, then it is the same as the limit you are looking for. So consider
                      $$
                      lim_{tto0^+}frac{1}{t^3}left(
                      sqrt{frac{1}{t^2}+sqrt{frac{1}{t^4}+1}}-frac{sqrt{2}}{t}
                      right)=
                      lim_{tto0^+}frac{sqrt{1+sqrt{1+t^4}}-sqrt{2}}{t^4}
                      $$
                      Now the dependency is only on $t^4$, so the limit is the same as
                      $$
                      lim_{uto0^+}frac{sqrt{1+sqrt{1+u}}-sqrt{2}}{u}
                      $$
                      which is the derivative at $0$ of $f(u)=sqrt{1+sqrt{1+u}}$.




                      Since $$f'(u)=frac{1}{2sqrt{1+sqrt{1+u}}}frac{1}{2sqrt{1+u}}$$ we have $$f'(0)=frac{1}{4sqrt{2}}$$







                      share|cite|improve this answer














                      Formally substitute $n=1/t$; if the function you get has a limit for $tto0^+$, then it is the same as the limit you are looking for. So consider
                      $$
                      lim_{tto0^+}frac{1}{t^3}left(
                      sqrt{frac{1}{t^2}+sqrt{frac{1}{t^4}+1}}-frac{sqrt{2}}{t}
                      right)=
                      lim_{tto0^+}frac{sqrt{1+sqrt{1+t^4}}-sqrt{2}}{t^4}
                      $$
                      Now the dependency is only on $t^4$, so the limit is the same as
                      $$
                      lim_{uto0^+}frac{sqrt{1+sqrt{1+u}}-sqrt{2}}{u}
                      $$
                      which is the derivative at $0$ of $f(u)=sqrt{1+sqrt{1+u}}$.




                      Since $$f'(u)=frac{1}{2sqrt{1+sqrt{1+u}}}frac{1}{2sqrt{1+u}}$$ we have $$f'(0)=frac{1}{4sqrt{2}}$$








                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited Dec 4 at 12:36

























                      answered Dec 4 at 10:28









                      egreg

                      176k1384198




                      176k1384198












                      • As always I don't agree that you give a full answer. In my opinion it is not useful for the asker nor for the site. I don't downvote because it is of course a correct and good answer. But virtually I give you a "-1".
                        – gimusi
                        Dec 4 at 10:31




















                      • As always I don't agree that you give a full answer. In my opinion it is not useful for the asker nor for the site. I don't downvote because it is of course a correct and good answer. But virtually I give you a "-1".
                        – gimusi
                        Dec 4 at 10:31


















                      As always I don't agree that you give a full answer. In my opinion it is not useful for the asker nor for the site. I don't downvote because it is of course a correct and good answer. But virtually I give you a "-1".
                      – gimusi
                      Dec 4 at 10:31






                      As always I don't agree that you give a full answer. In my opinion it is not useful for the asker nor for the site. I don't downvote because it is of course a correct and good answer. But virtually I give you a "-1".
                      – gimusi
                      Dec 4 at 10:31




















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