MaxLimit
MaxLimit[f,xx^{*}]
gives the max limit _{xx*}f(x).
MaxLimit[f,{x_{1},…,x_{n}}]
gives the nested max limit ⋯ f (x_{1},…,x_{n}).
MaxLimit[f,{x_{1},…,x_{n}}{,…,}]
gives the multivariate max limit f (x_{1},…,x_{n}).
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 realvalued 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,{x_{1},…,x_{n}}]  For a finite limit point x^{*} and {,…,}:

MaxLimit[f,xx^{*}]f^{*} MaxLimit[f,{x_{1},…,x_{n}}{,…,}]f^{*}  The definition uses the max envelope max[ϵ]==MaxValue[{f[x],0<<ϵ},x] for univariate f[x] and max[ϵ]==MaxValue[{f[x_{1},…,x_{n}],0<<ϵ},{x_{1},…,x_{n}}] for multivariate f[x_{1},…,x_{n}]. 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[x_{1},…,x_{n}],x_{1}>ω∧⋯∧x_{n}>ω},{x_{1},…,x_{n}}] 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[x_{1},x_{2}]] 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 {dir_{1},…,dir_{n}} use direction dir_{i} for variable x_{i} 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, MaxLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.
Examples
open allclose allBasic Examples (3)
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 Infinity:
The nested max limit as first and then :
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 :
TraditionalForm formatting:
Elementary Functions (10)
Piecewise Functions (5)
A discontinuous piecewise function:
A leftcontinuous piecewise function:
The twosided max limit is the larger of the two:
UnitStep is effectively a rightcontinuous 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)
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 twodimensional 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 twodimensional 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 twodimensional max limit at the origin is :
Reexpress 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:
Options (10)
Assumptions (1)
Specify conditions on parameters using Assumptions:
Direction (5)
The default direction is Reals:
"TwoSided" is equivalent to Reals:
Max limit in the complex plane:
Compare with the limit over the reals:
Compute the bivariate max limit approach from different quadrants:
Approaching the origin from the first quadrant:
Approaching the origin from the second quadrant:
Approaching the origin from the left halfplane:
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 GenerateConditions>True, even these nongeneric 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 (9)
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 :
In increasingly small regions around , gets increasingly flat and more of the graph is below :
The function does not have a limit as approaches :
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 "bigo of " at , written , if :
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 fifthorder Taylor polynomial at :
Find the motion of a critically driven massspring system:
The motion is oscillatory but becomes arbitrarily large, indicating an instability:
The oscillatory motion is bounded and eventually restricted to , indicating stability:
Differentiation (3)
The leftupper Dini derivative is defined as:
The rightupper 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 rightupper Dini derivative , defined as follows:
The rightlower 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 realvalued 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:
For a realvalued 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 the two max limits are equal—as in this example—then 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 realvalued functions: