AsymptoticEquivalent

AsymptoticEquivalent[f,g,xx*]

gives conditions for as xx*.

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

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

Details and Options

  • Asymptotic equivalent is also expressed as f is asymptotic to g and f is asymptotically equivalent to g. The point x* is often assumed from context.
  • Asymptotic equivalent is an equivalence relation and means TemplateBox[{{{f, (, x, )}, -, {g, (, x, )}}}, Abs]<=cTemplateBox[{{g, (, x, )}}, Abs] when x is near x* for all constants . It is a finer asymptotic equivalence relation than AsymptoticEqual.
  • Typical uses include simple expressions for functions and sequences near some point. It is frequently used for asymptotic solutions to equations.
  • For a finite limit point x* and {,,}:
  • AsymptoticEquivalent[f[x],g[x],xx*]for all there exists such that 0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<delta(c,x^*) implies TemplateBox[{{{f, (, x, )}, -, {g, (, x, )}}}, Abs]<=cTemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticEquivalent[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]for all there exists such that 0<TemplateBox[{{{, {{{x, _, 1}, -, {x, _, {(, 1, )}, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, {(, n, )}, ^, *}}}, }}}, Norm]<delta(epsilon,x^*) implies TemplateBox[{{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}, -, {g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}}, Abs]<=cTemplateBox[{{g, (, x, )}}, Abs]
  • For an infinite limit point:
  • AsymptoticEquivalent[f[x],g[x],x]for all there exists such that implies TemplateBox[{{{{f, (, x, )}, /, {g, (, x, )}}, -, 1}}, Abs]<=c
    AsymptoticEquivalent[f[x1,,xn],g[x1,,xn],{x1,,xn}{,,}]for all there exists such that implies TemplateBox[{{{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}, /, {g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, -, 1}}, Abs]<=c
  • AsymptoticEquivalent[f[x],g[x],xx*] exists if and only if Limit[f[x]/g[x],xx*]1 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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The ratio of the functions approaches as :

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Verify that as :

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The ratio of the functions approaches as and become large:

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Scope  (10)

Options  (9)

Applications  (12)

Properties & Relations  (4)

See Also

AsymptoticEqual  AsymptoticLess  AsymptoticGreater  AsymptoticLessEqual  AsymptoticGreaterEqual  Limit

Introduced in 2018
(11.3)