# MaxLimit

MaxLimit[f,xx*]

gives the max limit xx*f(x).

MaxLimit[f,{x1,,xn}]

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

MaxLimit[f,{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*]f* MaxLimit[f,{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:

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

An infinite max limit:

Close to the discontinuity, there are arbitrarily large values:

Max limit from above:

Max limit from below:

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

## Scope(35)

### Basic Uses(5)

Find the max limit at a point:

Find the max limit at a symbolic point:

Find the max limit at :

The nested max limit as first and then :

The nested max limit as and then :

Compute the multivariate max 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:

Logarithmic functions:

The function decays faster than any power of as :

Conversely, blows up faster than any power of :

Visualize representative functions:

### Piecewise Functions(5)

A discontinuous piecewise function:

A left-continuous piecewise function:

The two-sided max limit is the larger 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 max limit of Floor as x approaches integer values:

### Special Functions(4)

Upper limits involving Gamma:

Upper limits involving Bessel-type functions:

Upper limits involving exponential integrals:

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

### Nested Max Limits(3)

Compute the nested max limit as first and then :

The same result is obtained by computing two MaxLimit expressions:

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

This is again equivalent to two nested max limits:

The nested max limit as first and then is :

The nested max limit as first and then is :

Consider the function for two variables at the origin:

The iterated max limit as and then is :

The iterated max limit as and then is :

The true bivariate max limit is , as points where almost cancels give arbitrarily large values:

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

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

### Multivariate Max Limits(4)

Find the max limit of a multivariate function:

The two nested max limits give different answers:

Approaching the origin along the curve yields a third result:

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

This is achieved along the curve :

Visualize the minimum and maximum values near the origin:

Find the max limit of a bivariate function:

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

Note that neither iterated limit gives this result:

Indeed, along any rate the function is constant:

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

Visualize the function and the three max limits computed:

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

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

Re-express the function in terms of polar coordinates:

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

Compute the max limit of a trivariate function:

The max limit at the origin is :

Note that the various iterated max limits are 0:

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

The max 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)

Max limit from below:

Equivalently:

Max limit from above:

Equivalently:

The default direction is Reals:

"TwoSided" is equivalent to Reals:

Max limit in the complex plane:

Compare with the limit over the reals:

Max limits at a branch cut:

Compute the bivariate max 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(13)

### Geometry of Max Limits(3)

The function has a max 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 limit exists even though itself does not have a limit as :

The function has the limit zero as approaches :

Thus, its max limit is zero:

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

The function does not have a limit as approaches :

However, its max limit is :

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

### Asymptotic Analysis(3)

A function is said to be "big-o of " at , 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 at least as fast as :

From Taylor's theorem, if has continuous derivatives around , then :

This is the fifth-order Taylor polynomial at :

The definition of is :

Verify that :

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

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

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

### Continuity(4)

A function is upper semicontinuous at if . UnitStep is upper semicontinuous at the origin:

Visualize the function:

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

Visualize the function:

Consider the following function:

This function is upper semicontinuous at the origin:

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

Note that the MaxLimit of f does not depend on the value of f at zero, so any value greater than one would also make f upper semicontinuous:

Visualize f:

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

Floor is discontinuous but upper semicontinuous at every integer:

On the other hand, Ceiling is neither continuous nor upper semicontinuous at the integers:

Both are continuous at noninteger values, but only Floor is upper semicontinuous on all of :

### Differentiation(3)

The left-upper Dini derivative is defined as:

The right-upper Dini derivative is defined similarly:

Ramp has finite upper 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 growth of the function around zero is bounded:

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

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

is right 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 right derivative at the origin:

## Properties & Relations(13)

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

The corresponding limit does not exist:

Positive multiplicative constants can be moved outside a max limit:

If and have finite max limits as , then :

In this case, there is strict inequality:

Assumptions apply to parameters in the max limit expression:

Direction places conditions on the limit variable:

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

Compare with the following:

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

If has a finite limit as , then :

MaxLimit is always greater than or equal to MinLimit:

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

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

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

If for , then :

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

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

MaxLimit is always greater than or equal to DiscreteMaxLimit:

## Possible Issues(1)

MaxLimit is only defined for real-valued functions:

## Neat Examples(1)

Visualize a set of max limits:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_maxlimit, author="Wolfram Research", title="{MaxLimit}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/MaxLimit.html}", note=[Accessed: 15-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_maxlimit, organization={Wolfram Research}, title={MaxLimit}, year={2017}, url={https://reference.wolfram.com/language/ref/MaxLimit.html}, note=[Accessed: 15-August-2024 ]}