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 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],xx*] for all there exists such that implies AsymptoticEquivalent[f[x1,…,xn],g[x1,…,xn],{x1,…,xn}{,…,}] for all there exists such that implies
• For an infinite limit point:
•  AsymptoticEquivalent[f[x],g[x],x∞] for all there exists such that implies AsymptoticEquivalent[f[x1,…,xn],g[x1,…,xn],{x1,…,xn}{∞,…,∞}] for all there exists such that implies
• 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 \$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 nongeneric 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 :

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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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