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No output from Limit[ ]?

Posted 3 years ago

enter image description here

I'm a novice I want to ask why I can't find out the limit of input.

POSTED BY: yongqi yongqi
8 Replies

This type integral can't be found a symbolic solution - closed-form solution. With numerics is easy:

  f[n_?NumericQ] := 
   NIntegrate[Sqrt[n^2 - x^2]/(2 + x^(-x)), {x, 0, n}]; Table[
   f[n]/n^2 /. n -> 10^k, {k, 1, 6}]

 (*{0.359439, 0.389349, 0.392364, 0.392699, 0.392699, 0.392699}*)

Looks like limit have a value: 0.392699...

EDITED:

We can use:

 AsymptoticLessEqual[x^-x, 1/(x^2 + 1), x -> \[Infinity]]
 (*True*)

then:

 Limit[Integrate[
   Sqrt[n^2 - x^2]/(2 + (1/(x^2 + 1))) // Factor, {x, 0, n}, 
   Assumptions -> {n > 0}]/n^2, n -> Infinity]
(*\[Pi]/8*)

 \[Pi]/8 // N
 (*0.392699*)
POSTED BY: Mariusz Iwaniuk

Thank you, friends.

POSTED BY: yongqi yongqi
Posted 3 years ago

Please post code, not images.

Notice the ( ) are colored red to indicate a syntax error. In the WL arguments to functions are enclosed in [ ], not ( ).

POSTED BY: Rohit Namjoshi

Thank you very much.

POSTED BY: yongqi yongqi

I also want to ask why the calculation is so slow?

Does this software need to be connected to the cloud?

POSTED BY: yongqi yongqi
Posted 3 years ago

Slow compared to what?

Some integrals are easy to compute, some are not, and some do not have an analytic solution. Some limits are easy to compute, some are not.

POSTED BY: Rohit Namjoshi

OK, thank you.

POSTED BY: yongqi yongqi

Why does the calculation result just repeat the title, but not come to a specific result?

enter image description here

\(\*UnderscriptBox[\(\[Limit]\), \(n \[Rule] \[Infinity]\)]\)\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(n\)]\(
\*FractionBox[
SqrtBox[\(
\*SuperscriptBox[\(n\), \(2\)] - 
\*SuperscriptBox[\(x\), \(2\)]\)], \(2 + 
\*SuperscriptBox[\(x\), \(-x\)]\)] \[DifferentialD]x\)\)/n^2
POSTED BY: yongqi yongqi
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