MinLimit

MinLimit[f[x],xx*]

gives the min limit xx*f(x).

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

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

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

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

Details and Options

  • MinLimit is also known as limit inferior, infimum limit, liminf, lower limit and inner limit.
  • MinLimit computes the largest lower 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 \[MinLimit], with underscripts or subscripts, min limits can be entered as follows:
  • fmin limit in the default direction
    fmin limit from above
    fmin limit from below
    fmin limit in the complex plane
    fMinLimit[f,{x1,,xn}]
  • For a finite limit point x* and {,,}:
  • MinLimit[f[x],xx*]f*TemplateBox[{{min, (, 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^*
    MinLimit[f[x1,,xn],{x1,,xn}{,,}]f*TemplateBox[{{min, (, 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 min envelope min[ϵ]MinValue[{f[x],0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<ϵ},x] for univariate f[x] and min[ϵ]MinValue[{f[x1,,xn],0<TemplateBox[{{{, {{{x, _, {(, 1, )}}, -, {x, _, {(, 1, )}, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, {(, n, )}, ^, *}}}, }}}, Norm]<ϵ},{x1,,xn}] for multivariate f[x1,,xn]. The function min[ϵ] is monotone increasing as ϵ0, so it always has a limit, which may be ±.
  • The illustration shows min[TemplateBox[{{x, -, {x, ^, *}}}, Abs]] and min[] in blue.
  • For an infinite limit point x*, the min envelope min[ω]MinValue[{f[x],x>ω},x] is used for univariate f[x] and min[ω]MinValue[{f[x1,,xn],x1>ωxn>ω},{x1,,xn}] for multivariate f[x1,,xn]. The function min[ω] is monotone increasing as ω, so it always has a limit.
  • The illustration shows min[x] and min[Min[x1,x2]] in blue.
  • MinLimit returns unevaluated when the min 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, MinLimit 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 min 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 min limit:

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Close to the discontinuity, there are arbitrarily small values:

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Min limit from above:

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Min limit from below:

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The two-sided min limit is the smaller of the two:

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Scope  (35)

Options  (10)

Applications  (8)

Properties & Relations  (13)

Possible Issues  (1)

Neat Examples  (1)

Introduced in 2017
(11.2)