# MinLimit

MinLimit[f[x],xx*]

gives the min limit xx*f(x).

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

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

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

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

# Details and Options

• MinLimit is also known as limit inferior, infimum limit, liminf, lower limit and inner limit.
• MinLimit computes the largest lower 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 \[MinLimit], with underscripts or subscripts, min limits can be entered as follows:
•  f min limit in the default direction f min limit from above f min limit from below f min limit in the complex plane …f MinLimit[f,{x1,…,xn}]
• For a finite limit point x* and {,,}:
•  MinLimit[f[x],xx*]f* MinLimit[f[x1,…,xn],{x1,…,xn}{,…,}]f*
• The definition uses the min envelope min[ϵ]MinValue[{f[x],0<<ϵ},x] for univariate f[x] and min[ϵ]MinValue[{f[x1,,xn],0<<ϵ},{x1,,xn}] for multivariate f[x1,,xn]. The function min[ϵ] is monotone increasing as ϵ0, so it always has a limit, which may be ±.
• The illustration shows min[] and min[] in blue.
• For an infinite limit point x*, the min envelope min[ω]MinValue[{f[x],x>ω},x] is used for univariate f[x] and min[ω]MinValue[{f[x1,,xn],x1>ωxn>ω},{x1,,xn}] for multivariate f[x1,,xn]. The function min[ω] is monotone increasing as ω, so it always has a limit.
• The illustration shows min[x] and min[Min[x1,x2]] in blue.
• MinLimit returns unevaluated when the min 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, MinLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

# Examples

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## Basic Examples(3)

A min 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 min limit:

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Close to the discontinuity, there are arbitrarily small values:

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Min limit from above:

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Min limit from below:

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The two-sided min limit is the smaller of the two:

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