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


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


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[k],k-] is equivalent to DiscreteMaxLimit[f[-l],l] etc.
  • The definition uses the max envelope max[ω]MaxValue[{f[k],kωk},k] for univariate f[k] and max[ω]MaxValue[{f[k1,,kn],k1ωknωki},{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.


open all close all

Basic Examples  (4)

Max limit of a sequence:

Click for copyable input

Max limit of a product:

Click for copyable input
Click for copyable input

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

Click for copyable input

TraditionalForm typesetting:

Click for copyable input

Scope  (21)

Options  (6)

Applications  (7)

Properties & Relations  (11)

Possible Issues  (1)

Neat Examples  (1)

Introduced in 2017