How can I solve the limit by mathematica?

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I use the Limit and DiscreteLimit to solve it but failed. How to solve it by Mathematica?
$$undersetnto infty textlimn left(int_0^fracpi 4 tan ^nleft(fracxnright) , dxright)^1/n$$






calculus-and-analysis







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  • If you try Integrate[Tan[x/n]^n, x, 0, Pi/4] MMA evaluates a ConditionalExpression which constrains 0<n<1/2. Might be the limit doesn't exist?
    – Ulrich Neumann
    2 days ago






  • 1




    @UlrichNeumann I'm quite sure, it does exist. At n->Infinity we can probably replace Tan[x/n] with x/n, then the limit is solvable and gives Pi/4
    – LLlAMnYP
    2 days ago










  • @LLlAMnYP The limit would be 1/4 [Pi]^((4 + [Pi])/[Pi]) (4 + [Pi])^(-4/[Pi]) (not Pi/4). I only tried to indicate, that MMA can't integrate if n>1/2
    – Ulrich Neumann
    2 days ago







  • 1




    @King.Max as you found, as it is Mathematica doesn't solve the limit. It needs some help (i.e. manual interference from the user). If my suggestion isn't acceptable to you, what kind of help would be acceptable? Maybe one could pass some options to Limit or to Integrate, but I can't say for sure.
    – LLlAMnYP
    2 days ago






  • 1




    Try: Limit[n*Integrate[Tan[x/n]^n, x, 0, Pi/4, Assumptions -> n > 1]^(1/n) // PowerExpand, n -> Infinity]
    – Mariusz Iwaniuk
    2 days ago














up vote
3
down vote

favorite












I use the Limit and DiscreteLimit to solve it but failed. How to solve it by Mathematica?
$$undersetnto infty textlimn left(int_0^fracpi 4 tan ^nleft(fracxnright) , dxright)^1/n$$






calculus-and-analysis







share|improve this question







New contributor




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



















  • If you try Integrate[Tan[x/n]^n, x, 0, Pi/4] MMA evaluates a ConditionalExpression which constrains 0<n<1/2. Might be the limit doesn't exist?
    – Ulrich Neumann
    2 days ago






  • 1




    @UlrichNeumann I'm quite sure, it does exist. At n->Infinity we can probably replace Tan[x/n] with x/n, then the limit is solvable and gives Pi/4
    – LLlAMnYP
    2 days ago










  • @LLlAMnYP The limit would be 1/4 [Pi]^((4 + [Pi])/[Pi]) (4 + [Pi])^(-4/[Pi]) (not Pi/4). I only tried to indicate, that MMA can't integrate if n>1/2
    – Ulrich Neumann
    2 days ago







  • 1




    @King.Max as you found, as it is Mathematica doesn't solve the limit. It needs some help (i.e. manual interference from the user). If my suggestion isn't acceptable to you, what kind of help would be acceptable? Maybe one could pass some options to Limit or to Integrate, but I can't say for sure.
    – LLlAMnYP
    2 days ago






  • 1




    Try: Limit[n*Integrate[Tan[x/n]^n, x, 0, Pi/4, Assumptions -> n > 1]^(1/n) // PowerExpand, n -> Infinity]
    – Mariusz Iwaniuk
    2 days ago












up vote
3
down vote

favorite









up vote
3
down vote

favorite











I use the Limit and DiscreteLimit to solve it but failed. How to solve it by Mathematica?
$$undersetnto infty textlimn left(int_0^fracpi 4 tan ^nleft(fracxnright) , dxright)^1/n$$






calculus-and-analysis







share|improve this question







New contributor




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











I use the Limit and DiscreteLimit to solve it but failed. How to solve it by Mathematica?
$$undersetnto infty textlimn left(int_0^fracpi 4 tan ^nleft(fracxnright) , dxright)^1/n$$






calculus-and-analysis




calculus-and-analysis






share|improve this question







New contributor




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











share|improve this question







New contributor




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









share|improve this question




share|improve this question






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asked 2 days ago









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











  • If you try Integrate[Tan[x/n]^n, x, 0, Pi/4] MMA evaluates a ConditionalExpression which constrains 0<n<1/2. Might be the limit doesn't exist?
    – Ulrich Neumann
    2 days ago






  • 1




    @UlrichNeumann I'm quite sure, it does exist. At n->Infinity we can probably replace Tan[x/n] with x/n, then the limit is solvable and gives Pi/4
    – LLlAMnYP
    2 days ago










  • @LLlAMnYP The limit would be 1/4 [Pi]^((4 + [Pi])/[Pi]) (4 + [Pi])^(-4/[Pi]) (not Pi/4). I only tried to indicate, that MMA can't integrate if n>1/2
    – Ulrich Neumann
    2 days ago







  • 1




    @King.Max as you found, as it is Mathematica doesn't solve the limit. It needs some help (i.e. manual interference from the user). If my suggestion isn't acceptable to you, what kind of help would be acceptable? Maybe one could pass some options to Limit or to Integrate, but I can't say for sure.
    – LLlAMnYP
    2 days ago






  • 1




    Try: Limit[n*Integrate[Tan[x/n]^n, x, 0, Pi/4, Assumptions -> n > 1]^(1/n) // PowerExpand, n -> Infinity]
    – Mariusz Iwaniuk
    2 days ago
















  • If you try Integrate[Tan[x/n]^n, x, 0, Pi/4] MMA evaluates a ConditionalExpression which constrains 0<n<1/2. Might be the limit doesn't exist?
    – Ulrich Neumann
    2 days ago






  • 1




    @UlrichNeumann I'm quite sure, it does exist. At n->Infinity we can probably replace Tan[x/n] with x/n, then the limit is solvable and gives Pi/4
    – LLlAMnYP
    2 days ago










  • @LLlAMnYP The limit would be 1/4 [Pi]^((4 + [Pi])/[Pi]) (4 + [Pi])^(-4/[Pi]) (not Pi/4). I only tried to indicate, that MMA can't integrate if n>1/2
    – Ulrich Neumann
    2 days ago







  • 1




    @King.Max as you found, as it is Mathematica doesn't solve the limit. It needs some help (i.e. manual interference from the user). If my suggestion isn't acceptable to you, what kind of help would be acceptable? Maybe one could pass some options to Limit or to Integrate, but I can't say for sure.
    – LLlAMnYP
    2 days ago






  • 1




    Try: Limit[n*Integrate[Tan[x/n]^n, x, 0, Pi/4, Assumptions -> n > 1]^(1/n) // PowerExpand, n -> Infinity]
    – Mariusz Iwaniuk
    2 days ago















If you try Integrate[Tan[x/n]^n, x, 0, Pi/4] MMA evaluates a ConditionalExpression which constrains 0<n<1/2. Might be the limit doesn't exist?
– Ulrich Neumann
2 days ago




If you try Integrate[Tan[x/n]^n, x, 0, Pi/4] MMA evaluates a ConditionalExpression which constrains 0<n<1/2. Might be the limit doesn't exist?
– Ulrich Neumann
2 days ago




1




1




@UlrichNeumann I'm quite sure, it does exist. At n->Infinity we can probably replace Tan[x/n] with x/n, then the limit is solvable and gives Pi/4
– LLlAMnYP
2 days ago




@UlrichNeumann I'm quite sure, it does exist. At n->Infinity we can probably replace Tan[x/n] with x/n, then the limit is solvable and gives Pi/4
– LLlAMnYP
2 days ago












@LLlAMnYP The limit would be 1/4 [Pi]^((4 + [Pi])/[Pi]) (4 + [Pi])^(-4/[Pi]) (not Pi/4). I only tried to indicate, that MMA can't integrate if n>1/2
– Ulrich Neumann
2 days ago





@LLlAMnYP The limit would be 1/4 [Pi]^((4 + [Pi])/[Pi]) (4 + [Pi])^(-4/[Pi]) (not Pi/4). I only tried to indicate, that MMA can't integrate if n>1/2
– Ulrich Neumann
2 days ago





1




1




@King.Max as you found, as it is Mathematica doesn't solve the limit. It needs some help (i.e. manual interference from the user). If my suggestion isn't acceptable to you, what kind of help would be acceptable? Maybe one could pass some options to Limit or to Integrate, but I can't say for sure.
– LLlAMnYP
2 days ago




@King.Max as you found, as it is Mathematica doesn't solve the limit. It needs some help (i.e. manual interference from the user). If my suggestion isn't acceptable to you, what kind of help would be acceptable? Maybe one could pass some options to Limit or to Integrate, but I can't say for sure.
– LLlAMnYP
2 days ago




1




1




Try: Limit[n*Integrate[Tan[x/n]^n, x, 0, Pi/4, Assumptions -> n > 1]^(1/n) // PowerExpand, n -> Infinity]
– Mariusz Iwaniuk
2 days ago




Try: Limit[n*Integrate[Tan[x/n]^n, x, 0, Pi/4, Assumptions -> n > 1]^(1/n) // PowerExpand, n -> Infinity]
– Mariusz Iwaniuk
2 days ago










1 Answer
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Integrate returns an anti-derivative which seems reasonable for large n:



antiDeri = Integrate[Tan[x/n]^n, x];
Plot[antiDeri /. n -> 18, x, 0, π/4, PlotRange -> All]




And then



Limit[n Power[(antiDeri /. x -> π/4) - (antiDeri /. x -> 0), 1/n], n -> ∞]



π/4






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  • how simple a tricky solution can be...
    – Ulrich Neumann
    2 days ago










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1 Answer
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up vote
6
down vote













Integrate returns an anti-derivative which seems reasonable for large n:



antiDeri = Integrate[Tan[x/n]^n, x];
Plot[antiDeri /. n -> 18, x, 0, π/4, PlotRange -> All]




And then



Limit[n Power[(antiDeri /. x -> π/4) - (antiDeri /. x -> 0), 1/n], n -> ∞]



π/4






share|improve this answer




















  • how simple a tricky solution can be...
    – Ulrich Neumann
    2 days ago














up vote
6
down vote













Integrate returns an anti-derivative which seems reasonable for large n:



antiDeri = Integrate[Tan[x/n]^n, x];
Plot[antiDeri /. n -> 18, x, 0, π/4, PlotRange -> All]




And then



Limit[n Power[(antiDeri /. x -> π/4) - (antiDeri /. x -> 0), 1/n], n -> ∞]



π/4






share|improve this answer




















  • how simple a tricky solution can be...
    – Ulrich Neumann
    2 days ago












up vote
6
down vote










up vote
6
down vote









Integrate returns an anti-derivative which seems reasonable for large n:



antiDeri = Integrate[Tan[x/n]^n, x];
Plot[antiDeri /. n -> 18, x, 0, π/4, PlotRange -> All]




And then



Limit[n Power[(antiDeri /. x -> π/4) - (antiDeri /. x -> 0), 1/n], n -> ∞]



π/4






share|improve this answer












Integrate returns an anti-derivative which seems reasonable for large n:



antiDeri = Integrate[Tan[x/n]^n, x];
Plot[antiDeri /. n -> 18, x, 0, π/4, PlotRange -> All]




And then



Limit[n Power[(antiDeri /. x -> π/4) - (antiDeri /. x -> 0), 1/n], n -> ∞]



π/4







share|improve this answer












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answered 2 days ago









Coolwater

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  • how simple a tricky solution can be...
    – Ulrich Neumann
    2 days ago
















  • how simple a tricky solution can be...
    – Ulrich Neumann
    2 days ago















how simple a tricky solution can be...
– Ulrich Neumann
2 days ago




how simple a tricky solution can be...
– Ulrich Neumann
2 days ago










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









 

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