AsymptoticLessEqual

AsymptoticLessEqual[f,g,xx*]

gives conditions for or as xx*.

AsymptoticLessEqual[f,g,{x1,,xn}{,,}]

gives conditions for or as {x1,,xn}{,,}.

Details and Options

  • Asymptotic less or equal is also expressed as f is big-o of g, f is upper bounded by g, f is of order at most g, and f grows at most as fast as g. The point x* is often assumed from context.
  • Asymptotic less or equal is an order relation and means TemplateBox[{{f, (, x, )}}, Abs]<=c TemplateBox[{{g, (, x, )}}, Abs] when x is near x* for some constant .
  • Typical uses include expressing simple upper bounds for functions and sequences near some point. It is frequently used for solutions to equations and to give simple upper bounds for computational complexity.
  • For a finite limit point x* and {,,}:
  • AsymptoticLessEqual[f[x],g[x],xx*]there exist and such that 0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<delta(c,x^*) implies TemplateBox[{{f, (, x, )}}, Abs]<=c TemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticLessEqual[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]there exist and such that 0<TemplateBox[{{{, {{{x, _, 1}, -, {x, _, 1, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, n, ^, *}}}, }}}, Norm]<delta(c,x^*) implies TemplateBox[{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=c TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]
  • For an infinite limit point:
  • AsymptoticLessEqual[f[x],g[x],x]there exist and such that implies TemplateBox[{{f, (, x, )}}, Abs]<=c TemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticLessEqual[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]there exist and such that implies TemplateBox[{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=c TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]
  • AsymptoticLessEqual[f[x],g[x],xx*] exists if and only if MaxLimit[Abs[f[x]/g[x]],xx*]< when g[x] does not have an infinite set of zeros near x*.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    DirectionRealsdirection to approach the limit point
    GenerateConditionsAutomaticgenerate conditions for parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"what 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:
  • Automaticnongeneric 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, Limit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (2)

Verify that as :

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Visualize the two functions:

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Verify that as {x,y}->:

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Visualize the two functions:

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Scope  (9)

Options  (10)

Applications  (10)

Properties & Relations  (8)

See Also

AsymptoticLess  AsymptoticGreaterEqual  AsymptoticGreater  AsymptoticEqual  AsymptoticEquivalent  MaxLimit

Introduced in 2018
(11.3)