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

Posted 5 months ago

I try to get the step-by-step solution for a limit as follows, but failed:

In[2]:= WolframAlpha["Limit[((x + E^x)^(1/x) - E^2 + 3/2 E^2 x)/x^2, \
x -> 0]", IncludePods -> "Input", AppearanceElements -> {"Pods"}, 
PodStates -> {"Input__Show steps"}]

Out[2]= \!\(
NamespaceBox["WolframAlphaQueryResults",
DynamicModuleBox[{Typeset`q$$ = 
"Limit[((x + E^x)^(1/x) - E^2 + 3/2 E^2 x)/x^2, x -> 0]", 
Typeset`opts$$ = {AppearanceElements -> {"Pods"}, 
IncludePods -> "Input", PodStates -> {"Input__Show steps"}}, 
Typeset`elements$$ = {"Pods"}, Typeset`asyncpods$$ = {}, 
Typeset`nonpods$$ = {}, Typeset`initdone$$ = 
FrontEnd`SuppressEvaluationHash[True], 
Typeset`queryinfo$$ = {"success" -> "true", 
"error" -> "false", {"http://www.w3.org/XML/1998/namespace", 
"space"} -> "preserve", "numpods" -> "0", "datatypes" -> "", 
"timedout" -> "", "timedoutpods" -> "", "timing" -> "1.383", 
"parsetiming" -> "1.0210000000000001", 
"parsetimedout" -> "false", "recalculate" -> "", 
"id" -> "MSP1075514e8ee62h3i906h400004d0123254hh0107g", 
"host" -> "https://www6b3.wolframalpha.com", "server" -> "16", 
"related" -> "https://www6b3.wolframalpha.com/api/v1/\
relatedQueries.jsp?id=\
MSPa1075614e8ee62h3i906h400001abaie5gdab5b1ib8350669649384844499", 
"version" -> "2.6", 
"inputstring" -> "Limit[((x + E^x)^(1/x) - E^2 + 3/2 E^2 x)/x^2, \
x -> 0]"}, 
Typeset`sessioninfo$$ = {"TimeZone" -> 8., 
"Date" -> {2023, 10, 10, 9, 11, 16.72927`7.976051975256879}, 
"Line" -> 2, "SessionID" -> 24217400133690190606}, 
Typeset`showpods$$ = {}, Typeset`failedpods$$ = {}, 
Typeset`chosen$$ = {}, Typeset`open$$ = False, Typeset`newq$$ = 
"Limit[((x + E^x)^(1/x) - E^2 + 3/2 E^2 x)/x^2, x -> 0]"}, 
DynamicBox[ToBoxes[
AlphaIntegration`FormatAlphaResults[
Dynamic[{1, {}, {}, Typeset`chosen$$, Typeset`open$$, 
Typeset`elements$$, Typeset`q$$, Typeset`opts$$, 
Typeset`nonpods$$, Typeset`queryinfo$$, 
Typeset`sessioninfo$$, Typeset`showpods$$, 
Typeset`failedpods$$, Typeset`newq$$}]], StandardForm],
ImageSizeCache->{20., {11.1875, 16.8125}},
TrackedSymbols:>{Typeset`showpods$$, Typeset`failedpods$$}],
DynamicModuleValues:>{},
Initialization:>If[
And[
Not[Typeset`initdone$$ === True], 
Not[Typeset`initdone$$ === FrontEnd`SuppressEvaluationHash[True]]], 
Null; WolframAlphaClient`Private`doAsyncUpdates[
Hold[{}], Typeset`asyncpods$$, 
Dynamic[Typeset`failedpods$$]]; Typeset`asyncpods$$ = {}; Typeset`\
initdone$$ = FrontEnd`SuppressEvaluationHash[True]],
SynchronousInitialization->False],
BaseStyle->{Deployed -> True},
DeleteWithContents->True,
Editable->False,
SelectWithContents->True]\)

Any tips for this problem will be helpful.

Regards,
Zhao

POSTED BY: Hongyi Zhao
5 Replies

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
Posted 5 months 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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