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MinLimit[f,xx*]

gives the min limit xx*f(x).

MinLimit[f,{x1,…,xn}]

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

MinLimit[f,{x1,…,xn}{,…,}]

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

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Typeset Limits  
Elementary Functions  
Show More Show More
Piecewise Functions  
Special Functions  
Nested Min Limits  
Multivariate Min Limits  
Options  
Assumptions  
Direction  
GenerateConditions  
PerformanceGoal  
Applications  
Geometry of Min Limits  
Asymptotic Analysis  
Continuity  
Differentiation  
Properties & Relations  
Possible Issues  
Neat Examples  
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MinLimit

MinLimit[f,xx*]

gives the min limit xx*f(x).

MinLimit[f,{x1,…,xn}]

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

MinLimit[f,{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,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}{,…,}]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>ω∧⋯∧xn>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:

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

An infinite min limit:

Close to the discontinuity, there are arbitrarily small values:

Min limit from above:

Min limit from below:

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

Scope  (35)

Basic Uses  (5)

Find the min limit at a point:

Find the min limit at a symbolic point:

Find the min limit at -Infinity:

The nested min limit as first and then :

The nested min limit as and then :

Compute the multivariate min limit as :

Typeset Limits  (4)

Use mlim to enter the character, and to create an underscript:

Take a limit from above or below by using a superscript or on the limit point:

After typing zero, use to create a superscript:

To specify a direction of Reals or Complexes, enter the domain as an underscript on the character:

Enter the rule as ->, use to create an underscript, and type reals to enter :

TraditionalForm formatting:

Elementary Functions  (10)

Polynomials:

Rational functions at singular points:

Rational functions at ±Infinity:

Algebraic functions:

Trigonometric functions at singular points:

Trigonometric functions at ±Infinity:

Inverse trigonometric functions:

Exponential functions:

The function decays faster than any power of as :

Conversely, blows up faster than any power of , but the sign of the product depends on the parity of :

Visualize representative functions:

Logarithmic functions:

Piecewise Functions  (5)

A discontinuous piecewise function:

A left-continuous piecewise function:

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

UnitStep is effectively a right-continuous piecewise function:

RealSign is effectively a discontinuous piecewise function:

Note that TemplateBox[{0}, RealSign] is related to neither value:

Find the min limit of Floor as x approaches integer values:

Special Functions  (4)

Min limits involving Gamma:

Min limits involving Bessel-type functions:

Min limits involving exponential integrals:

At every non-positive even integer, Gamma diverges to from one side:

Nested Min Limits  (3)

Compute the nested min limit as first and then :

The same result is obtained by computing two MinLimit expressions:

Computing the min limit as first and then