MinLimit

MinLimit[f,xx*]

gives the min limit xx*f(x).

MinLimit[f,{x1,,xn}]

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

MinLimit[f,{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*]f* MinLimit[f,{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

open allclose all

Basic Examples(3)

A min limit at infinity:

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

An infinite min limit:

Close to the discontinuity, there are arbitrarily small values:

Min limit from above:

Min limit from below:

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

Scope(35)

Basic Uses(5)

Find the min limit at a point:

Find the min limit at a symbolic point:

Find the min limit at :

The nested min limit as first and then :

The nested min limit as and then :

Compute the multivariate min limit as :

Typeset Limits(4)

Use mlim to enter the character, and to create an underscript:

Take a limit from above or below by using a superscript or on the limit point:

After typing zero, use to create a superscript:

To specify a direction of Reals or Complexes, enter the domain as an underscript on the character:

Enter the rule as ->, use to create an underscript, and type reals to enter :

Elementary Functions(10)

Polynomials:

Rational functions at singular points:

Rational functions at :

Algebraic functions:

Trigonometric functions at singular points:

Trigonometric functions at :

Inverse trigonometric functions:

Exponential functions:

The function decays faster than any power of as :

Conversely, blows up faster than any power of , but the sign of the product depends on the parity of :

Visualize representative functions:

Logarithmic functions:

Piecewise Functions(5)

A discontinuous piecewise function:

A left-continuous piecewise function:

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

UnitStep is effectively a right-continuous piecewise function:

RealSign is effectively a discontinuous piecewise function:

Note that is related to neither value:

Find the min limit of Floor as x approaches integer values:

Special Functions(4)

Min limits involving Gamma:

Min limits involving Bessel-type functions:

Min limits involving exponential integrals:

At every non-positive even integer, Gamma diverges to from one side:

Nested Min Limits(3)

Compute the nested min limit as first and then :

The same result is obtained by computing two MinLimit expressions:

Computing the min limit as first and then yields a different answer:

This is again equivalent to two nested min limits:

The nested min limit as first and then is :

The nested min limit as first and then is :

Consider the function for two variables at the origin:

The iterated min limit as and then is :

The iterated min limit as and then is :

The true bivariate min limit is , as points where almost cancels give arbitrarily small values:

For example, this value can be approached along the curve :

Visualize the function and the values along the two axes computed previously:

Multivariate Min Limits(4)

Find the min limit of a multivariate function:

The two nested min limits give different answers:

Approaching the origin along the curves yields a third result:

The true two-dimensional lower limit of the function is , achieved along the axis:

Visualize the minimum and maximum values near the origin:

Find the min limit of a bivariate function:

The true two-dimensional min limit of the function is :

Note that neither iterated limit gives this result:

Indeed, along any rate , the function is constant:

The minimum is approached along curves with close to , such as :

Visualize the function and the three min limits computed:

Find the min limit of a bivariate function at the origin:

The true two-dimensional min limit at the origin is :

Re-express the function in terms of polar coordinates:

The polar expression is bounded and disappears as , leaving the min limit of Sin:

Compute the min limit of a trivariate function:

The min limit at the origin is :

Note that the various iterated min limits are 0:

This is because the minimum is achieved along the line , :

The min limit can also be understood by transforming to spherical coordinates:

Visualize the function:

Options(10)

Assumptions(1)

Specify conditions on parameters using Assumptions:

Different assumptions can produce different results:

Direction(5)

Min limit from below:

Equivalently:

Min limit from above:

Equivalently:

The default direction is Reals:

"TwoSided" is equivalent to Reals:

Min limit in the complex plane:

Compare with the limit over the reals:

Min limits at a branch cut:

Compute the bivariate min limit approach from different quadrants:

Approaching the origin from the first quadrant:

Equivalently:

Approaching the origin from the second quadrant:

Approaching the origin from the left half-plane:

Approaching the origin from the bottom half-plane:

Visualize the function:

GenerateConditions(3)

Return a result without stating conditions:

This result is only valid if n>0:

Return unevaluated if the results depend on the value of parameters:

By default, conditions are generated that return a unique result:

By default, conditions are not generated if only special values invalidate the result:

With , even these non-generic conditions are reported:

PerformanceGoal(1)

Use PerformanceGoal to avoid potentially expensive computations:

The default setting uses all available techniques to try to produce a result:

Applications(12)

Geometry of Min Limits(3)

The function has a min limit of at :

This means there must be a sequence for which as ; for example, :

Numerically, and quite quickly:

Compute the two sequence limits exactly:

Note that this sequence exists even though itself does not have a limit as :

The function has the limit zero as approaches :

Thus, its min limit is zero:

In increasingly small regions around , gets increasingly flat, and more of the graph is above :

The function does not have a limit as approaches :

However, its min limit is :

In increasingly small regions around , bounces wildly, but becomes a better and better floor for it:

Asymptotic Analysis(2)

A function is said to be "big-omega of " at a, written , if :

For example, is :

But it is not :

The statement is always true:

If and , then :

It is possible for functions to share neither relationship:

Thus, is a reflexive partial order on functions:

if goes to zero no faster than :

Find the motion of a critically driven mass-spring system:

The motion is oscillatory but becomes arbitrarily negative, indicating an instability:

The oscillatory motion is bounded and eventually restricted to , indicating stability:

Continuity(4)

A function is lower semicontinuous at t if . SawtoothWave is upper semicontinuous at :

Visualize the function:

On the other hand, RealSign is not lower semicontinuous at the origin:

Visualize the function:

Consider the following function:

This function is lower semicontinuous at the origin:

This is despite f having neither a left nor a right limit at the origin:

Note that the MinLimit of f does not depend on the value of f at zero, so any value less than would also make f lower semicontinuous:

Visualize f:

A function is upper semicontinuous at if . A real-valued function is continuous iff it is both upper and lower semicontinuous. UnitStep is upper semicontinuous at :

However, it is not upper semicontinuous, so it is discontinuous at the origin:

On the other hand, the following shows that TriangleWave is continuous at the origin:

Visualize the two functions:

Ceiling is discontinuous but lower semicontinuous at every integer:

On the other hand, Floor is neither continuous nor lower semicontinuous at the integers:

Both are continuous at noninteger values, but only Ceiling is lower semicontinuous on all of :

Differentiation(3)

The left-lower Dini derivative is defined as:

The right-lower Dini derivative is defined similarly:

Ramp has finite lower Dini derivatives on the whole real line:

Note that these two derivatives are equal everywhere except the origin:

This is a reflection of the fact that Ramp is differentiable everywhere except the origin:

Consider the following function:

It is continuous at the origin:

But it has neither a left nor a right derivative:

It does, however, have finite Dini derivatives:

This indicates that the decrease of the function around zero is bounded:

There are two left Dini derivatives. The first is the left-lower Dini derivative , defined as follows:

The right-lower Dini derivative is defined similarly using a max limit:

is left differentiable at if and only if the two are equal and finite, as in the case of Ramp at :

However, the function does not have a left derivative at the origin:

Properties & Relations(13)

A real-valued function always has a (possibly infinite) min limit:

The corresponding limit may not exist:

Positive multiplicative constants can be moved outside a min limit:

If and have finite min limits as , then :

In this case, there is strict inequality:

Assumptions applies to parameters in the min limit expression:

Direction places conditions on the limit variable:

When computing nested min limits, appropriate assumptions are generated on later limit variables:

Compare with the following:

For a real-valued function, if Limit exists, then MinLimit has the same value:

If has a finite limit as , then :

MinLimit is always less than or equal to MaxLimit:

If MinLimit and MaxLimit are equal, then the limit exists and equals their common value:

If the min limit is , then the max limit and thus the limit are also :

MinLimit can be computed as -MaxLimit[-f,]:

If for , then :

If the two min limits are equalas in this examplethen f has a limit as :

This is a generalization of the "squeezing" or "sandwich" theorem:

MinLimit is always less than or equal to DiscreteMinLimit:

Possible Issues(1)

MinLimit is only defined for real-valued functions:

Neat Examples(1)

Visualize a set of min limits:

Wolfram Research (2017), MinLimit, Wolfram Language function, https://reference.wolfram.com/language/ref/MinLimit.html.

Text

Wolfram Research (2017), MinLimit, Wolfram Language function, https://reference.wolfram.com/language/ref/MinLimit.html.

CMS

Wolfram Language. 2017. "MinLimit." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MinLimit.html.

APA

Wolfram Language. (2017). MinLimit. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MinLimit.html

BibTeX

@misc{reference.wolfram_2024_minlimit, author="Wolfram Research", title="{MinLimit}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/MinLimit.html}", note=[Accessed: 04-August-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_minlimit, organization={Wolfram Research}, title={MinLimit}, year={2017}, url={https://reference.wolfram.com/language/ref/MinLimit.html}, note=[Accessed: 04-August-2024 ]}