# MaxLimit

MaxLimit[f[x],xx*]

gives the max limit xx*f(x).

MaxLimit[f[x1,,xn],{x1,,xn}]

gives the nested max limit f (x1,,xn).

MaxLimit[f[x1,,xn],{x1,,xn}{,,}]

gives the multivariate max limit f (x1,,xn).

# Details and Options

• MaxLimit is also known as limit superior, supremum limit, limsup, upper limit and outer limit.
• MaxLimit computes the smallest upper bound for the limit and is always defined for real-valued functions. It is often used to give conditions of convergence and other asymptotic properties where no actual limit is needed.
• By using the character , entered as Mlim or \[MaxLimit], with underscripts or subscripts max limits can be entered as follows:
•  f max limit in the default direction f max limit from above f max limit from below f max limit in the complex plane …f MaxLimit[f,{x1,…,xn}]
• For a finite limit point x* and {,,}:
•  MaxLimit[f[x],xx*]f* MaxLimit[f[x1,…,xn],{x1,…,xn}{,…,}]f*
• The definition uses the max envelope max[ϵ]==MaxValue[{f[x],0<<ϵ},x] for univariate f[x] and max[ϵ]==MaxValue[{f[x1,,xn],0<<ϵ},{x1,,xn}] for multivariate f[x1,,xn]. The function max[ϵ] is monotone decreasing as ϵ0, so it always has a limit, which may be ±.
• The illustration shows max[] and max[] in blue.
• For an infinite limit point x*, the max envelope max[ω]MaxValue[{f[x],x>ω},x] is used for univariate f and max[ω]MaxValue[{f[x1,,xn],x1>ωxn>ω},{x1,,xn}] for multivariate f. The function max[ω] is monotone decreasing as ω, so it always has a limit.
• The illustration shows max[x] and max[Min[x1,x2]] in blue.
• MaxLimit returns unevaluated when the max limit cannot be found.
• The following options can be given:
•  Assumptions \$Assumptions assumptions on parameters Direction Reals directions to approach the limit point GenerateConditions Automatic whether to generate conditions on parameters Method Automatic method to use PerformanceGoal "Quality" aspects of performance 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, MaxLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

# Examples

open allclose all

## Basic Examples(3)

A max limit at infinity:

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The function gets closer and closer to 1 without ever touching it:

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An infinite max limit:

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Close to the discontinuity, there are arbitrarily large values:

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Max limit from above:

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Max limit from below:

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The two-sided max limit is the larger of the two:

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