MaxLimit

MaxLimit[f[x],xx*]

gives the max limit xx*f(x).

MaxLimit[f[x1,,xn],{x1,,xn}]

gives the nested max limit f (x1,,xn).

MaxLimit[f[x1,,xn],{x1,,xn}{,,}]

gives the multivariate max limit f (x1,,xn).

Details and Options

  • MaxLimit is also known as limit superior, supremum limit, limsup, upper limit and outer limit.
  • MaxLimit computes the smallest upper bound for the limit and is always defined for real-valued functions. It is often used to give conditions of convergence and other asymptotic properties where no actual limit is needed.
  • By using the character , entered as Mlim or \[MaxLimit], with underscripts or subscripts max limits can be entered as follows:
  • fmax limit in the default direction
    fmax limit from above
    fmax limit from below
    fmax limit in the complex plane
    fMaxLimit[f,{x1,,xn}]
  • For a finite limit point x* and {,,}:
  • MaxLimit[f[x],xx*]f*TemplateBox[{{max, (, epsilon, )}, epsilon, 0, +, {Direction, ->, {-, 1}}}, LimitWithSuperscript, DisplayFunction -> ({Sequence[{Sequence["lim"], _, DocumentationBuild`Utils`Private`Parenth[{#2, ->, {#3, ^, DocumentationBuild`Utils`Private`Parenth[#4]}}, LimitsPositioning -> True]}], #1} & ), InterpretationFunction -> ({Limit, [, {#1, ,, {#2, ->, #3}, ,, #5}, ]} & )]=f^*
    MaxLimit[f[x1,,xn],{x1,,xn}{,,}]f*TemplateBox[{{max, (, epsilon, )}, epsilon, 0, +, {Direction, ->, {-, 1}}}, LimitWithSuperscript, DisplayFunction -> ({Sequence[{Sequence["lim"], _, DocumentationBuild`Utils`Private`Parenth[{#2, ->, {#3, ^, DocumentationBuild`Utils`Private`Parenth[#4]}}, LimitsPositioning -> True]}], #1} & ), InterpretationFunction -> ({Limit, [, {#1, ,, {#2, ->, #3}, ,, #5}, ]} & )]=f^*
  • The definition uses the max envelope max[ϵ]==MaxValue[{f[x],0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<ϵ},x] for univariate f[x] and max[ϵ]==MaxValue[{f[x1,,xn],0<TemplateBox[{{{, {{{x, _, {(, 1, )}}, -, {x, _, {(, 1, )}, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, {(, n, )}, ^, *}}}, }}}, Norm]<ϵ},{x1,,xn}] for multivariate f[x1,,xn]. The function max[ϵ] is monotone decreasing as ϵ0, so it always has a limit, which may be ±.
  • The illustration shows max[TemplateBox[{{x, -, {x, ^, *}}}, Abs]] and max[] in blue.
  • For an infinite limit point x*, the max envelope max[ω]MaxValue[{f[x],x>ω},x] is used for univariate f and max[ω]MaxValue[{f[x1,,xn],x1>ωxn>ω},{x1,,xn}] for multivariate f. The function max[ω] is monotone decreasing as ω, so it always has a limit.
  • The illustration shows max[x] and max[Min[x1,x2]] in blue.
  • MaxLimit returns unevaluated when the max limit cannot be found.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    DirectionRealsdirections to approach the limit point
    GenerateConditionsAutomaticwhether to generate conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"aspects of performance to optimize
  • Possible settings for Direction include:
  • Reals or "TwoSided"from both real directions
    "FromAbove" or -1from above or larger values
    "FromBelow" or +1from below or smaller values
    Complexesfrom all complex directions
    Exp[ θ]in the direction
    {dir1,,dirn}use direction diri for variable xi independently
  • DirectionExp[ θ] at x* indicates the direction tangent of a curve approaching the limit point x*.
  • Possible settings for GenerateConditions include:
  • Automaticnon-generic conditions only
    Trueall conditions
    Falseno conditions
    Nonereturn unevaluated if conditions are needed
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, MaxLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (3)

A max limit at infinity:

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The function gets closer and closer to 1 without ever touching it:

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An infinite max limit:

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Close to the discontinuity, there are arbitrarily large values:

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Max limit from above:

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Max limit from below:

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The two-sided max limit is the larger of the two:

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Scope  (35)

Options  (10)

Applications  (9)

Properties & Relations  (13)

Possible Issues  (1)

Neat Examples  (1)

Introduced in 2017
(11.2)