# 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 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],xx*] there exist and such that implies AsymptoticLessEqual[f[x1,…,xn],g[x1,…,xn],{x1,…,xn}{ ,…, }] there exist and such that implies • • For an infinite limit point:
•  AsymptoticLessEqual[f[x],g[x],x∞] there exist and such that implies AsymptoticLessEqual[f[x1,…,xn],g[x1,…,xn],{x1,…,xn}{∞,…,∞}] there exist and such that implies • • 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 \$Assumptions assumptions on parameters Direction Reals direction to approach the limit point GenerateConditions Automatic generate conditions for parameters Method Automatic method to use PerformanceGoal "Quality" what to optimize
• Possible settings for Direction include:
•  Reals or "TwoSided" from both real directions "FromAbove" or -1 from above or larger values "FromBelow" or +1 from below or smaller values Complexes from 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:
•  Automatic non-generic conditions only True all conditions False no conditions None return 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 :

 In:= Out= Visualize the two functions:

 In:= Out= Verify that as {x,y}->:

 In:= Out= Visualize the two functions:

 In:= Out= ## Properties & Relations(8)

Introduced in 2018
(11.3)