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Step-by-step solution for a limit failed to work

Hongyi Zhao
Posted 2 years ago
POSTED BY: Hongyi Zhao
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I do not know whether the step-by-step should work on this. One reasonable way to see why the limit is as claimed is to expand (x + E^x)^(1/x) as a series of order 1 at the origin.
POSTED BY: Daniel Lichtblau
Hongyi Zhao
Hongyi Zhao
Posted 2 years ago

What I want to know is how to do it manually and step-by-step.
POSTED BY: Hongyi Zhao

Sorry, I tried again now, and the limit is correct. I don't know what happened.
POSTED BY: Gianluca Gorni

Very strange. In Mathematica 13.3 the following

Limit[((x + E^x)^(1/x) - E^2 + 3/2 E^2 x)/x^2,
  x -> 0]

gives Indeterminate, contradicted by

Asymptotic[((x + E^x)^(1/x) - E^2 + 3/2 E^2 x)/x^2,
  x -> 0]
ComplexPlot3D[((x + E^x)^(1/x) - E^2 + 3/2 E^2 x)/x^2,
 {x, (-1 - I)/3, (1 + I)/3}]
POSTED BY: Gianluca Gorni

I tried in Mathematica 13.3.0 on Windows 10 give me:

 Limit[((x + E^x)^(1/x) - E^2 + 3/2 E^2 x)/x^2, x -> 0]
 (*(71 E^2)/24*)

Regards

POSTED BY: Mariusz Iwaniuk
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