DiscreteMinLimit

DiscreteMinLimit[f[k],k]

gives the min limit kf(k) of the sequence f[k] as k tends to over the integers.

DiscreteMinLimit[f[k1,,kn],{k1,,kn}]

gives the nested min limit f(k1,,kn) over the integers.

DiscreteMinLimit[f[k1,,kn],{k1,,kn}{,,}]

gives the multivariate min limit f(k1,,kn) over the integers.

Details and Options

  • DiscreteMinLimit is also known as limit inferior, infimum limit, liminf, lower limit and inner limit.
  • DiscreteMinLimit computes the largest lower bound for the limit and is always defined for real-valued sequences. It is often used to give conditions of convergence and other asymptotic properties that do not rely on an actual limit to exist.
  • DiscreteMinLimit[f,k] can be entered as f. A template can be entered as dmlim, and moves the cursor from the underscript to the body.
  • DiscreteMinLimit[f,{k1,,kn}{,,}] can be entered asf.
  • The possible limit points are ±.
  • The min limit is defined as a limit of the min envelope sequence min[ω]:
  • DiscreteMinLimit[f[k],k]DiscreteLimit[min[ω],ω]
    DiscreteMinLimit[f[k1,,kn],{k1,,kn}{,,}]DiscreteLimit[min[ω],ω]
  • DiscreteMinLimit[f[k],k-] is equivalent to DiscreteMinLimit[f[-l],l] etc.
  • The definition uses the min envelope min[ω]MinValue[{f[k],kωk},k] for univariate f[k] and min[ω]MinValue[{f[k1,,kn],k1ωknωki},{k1,,kn}] for multivariate f[k1,,kn]. The sequence min[ω] is monotone increasing as ω, so always has a limit, which may be ±.
  • The illustration shows min[k] and min[Min[k1,k2]] in blue.
  • DiscreteMinLimit returns unevaluated when the min limit cannot be found.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    GenerateConditionsAutomaticwhether to generate conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"aspects of performance to optimize
  • 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, DiscreteMinLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (4)

Min limit of a sequence:

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Min limit of a product:

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Use dmlim to enter the template and to move from the underscript to the body:

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TraditionalForm typesetting:

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Scope  (22)

Options  (6)

Applications  (3)

Properties & Relations  (11)

Possible Issues  (1)

Neat Examples  (1)

See Also

DiscreteMaxLimit  DiscreteLimit  MinLimit  MinValue  Min

Introduced in 2017
(11.2)