DiscreteLimit

DiscreteLimit[f[k],k]

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

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

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

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

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

Details and Options

  • DiscreteLimit is also known as discrete limit or limit over the integers.
  • DiscreteLimit computes the limiting value of a sequence f as its variables k or ki get arbitrarily large.
  • DiscreteLimit[f,k] can be entered as f. A template can be entered as dlim, and moves the cursor from the underscript to the body.
  • DiscreteLimit[f,{k1,,kn}{,,}] can be entered as f.
  • The possible limit points are ±.
  • For a finite limit value f*:
  • DiscreteLimit[f[k],k]f*for every there is a such that implies TemplateBox[{{{f, (, k, )}, -, {f, ^, *}}}, Abs]<epsilon
    DiscreteLimit[f[k1,,kn],{k1,,kn}{,,}]f*for every there is a such that implies TemplateBox[{{{f, (, {{k, _, 1}, ,, ..., ,, {k, _, n}}, )}, -, {f, ^, *}}}, Abs]<epsilon
  • DiscreteLimit[f[k],k-] is equivalent to DiscreteLimit[f[-l],l] etc.
  • DiscreteLimit returns Indeterminate when it can prove that the limit does not exist, and returns unevaluated when no limit can 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, DiscreteLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (4)

Limit of a sequence:

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Plot the sequence and its limit:

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Limit of a multivariate sequence:

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Plot the sequence and its limit:

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

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TraditionalForm typesetting:

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Scope  (37)

Options  (6)

Applications  (35)

Properties & Relations  (15)

Neat Examples  (1)

See Also

Limit  DiscreteMaxLimit  DiscreteMinLimit  SumConvergence  RSolveValue

Introduced in 2017
(11.2)