DiscreteMaxLimit

DiscreteMaxLimit[f[k],k∞]

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

DiscreteMaxLimit[f[k1,…,kn],{k1,…,kn}]

gives the nested max limit ⋯ f(k1,…,kn) over the integers.

DiscreteMaxLimit[f,{k1,…,kn}{,…,}]

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

Details and Options

  • DiscreteMaxLimit is also known as limit superior, supremum limit, limsup, upper limit and outer limit.
  • DiscreteMaxLimit computes the smallest upper 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.
  • DiscreteMaxLimit[f,k∞] can be entered as f. A template can be entered as dMlim, and moves the cursor from the underscript to the body.
  • DiscreteMaxLimit[f,{k1,…,kn}{,…,}] can be entered as …f.
  • The possible limit points are ±∞.
  • The max limit is defined as a limit of the max envelope sequence max[ω]:
  • DiscreteMaxLimit[f[k],k∞]DiscreteLimit[max[ω],ω∞]
    DiscreteMaxLimit[f[k1,…,kn],{k1,…,kn}{∞,…,∞}]DiscreteLimit[max[ω],ω∞]
  • DiscreteMaxLimit[f[k],k-∞] is equivalent to DiscreteMaxLimit[f[-l],l∞] etc.
  • The definition uses the max envelope max[ω]MaxValue[{f[k],k≥ω∧k∈},k] for univariate f[k] and max[ω]MaxValue[{f[k1,…,kn],k1≥ω∧⋯∧kn≥ω∧ki∈},{k1,…,kn}] for multivariate f[k1,…,kn]. The sequence max[ω] is monotone decreasing as ω∞, so it always has a limit, which may be ±∞.
  • The illustration shows max[k] and max[Min[k1,k2]] in blue.
  • DiscreteMaxLimit returns unevaluated when the max 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, DiscreteMaxLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (4)

Max limit of a sequence:

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Max 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  (21)

Options  (6)

Applications  (7)

Properties & Relations  (11)

Possible Issues  (1)

Neat Examples  (1)

Introduced in 2017
(11.2)