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 :

The nested max limit as and then :

Compute the multivariate max limit as :

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:

Polynomials:

Rational functions at singular points:

Rational functions at ±Infinity :

Algebraic functions:

Trigonometric functions at singular points:

Trigonometric functions at ±Infinity :

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:

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 TemplateBox[{0}, RealSign] is related to neither value:

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

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:

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:

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:

Specify conditions on parameters using Assumptions :

Different assumptions can produce different results:

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:

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 non-generic conditions are reported:

Use PerformanceGoal to avoid potentially expensive computations:

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

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:

A function is said to be "big-o of " at , written , if _(x->_(TemplateBox[{}, Reals])a)TemplateBox[{{{(, {f, (, x, )}, )}, /, {(, {g, (, x, )}, )}}}, Abs]<infty:

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 _(x->_(TemplateBox[{}, Reals])a)TemplateBox[{{{(, {f, (, x, )}, )}, /, {(, {g, (, x, )}, )}}}, Abs]<infty:

Verify that :

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

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

Add overdamping the system:

The oscillatory motion is bounded and eventually restricted to +/-(TemplateBox[{alpha}, RealAbs])/betasqrt(k/m), indicating stability:

A function is upper semicontinuous at if TemplateBox[{{f, (, x, )}, x, a}, MaxLimit2Arg]<=f(a). 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 TemplateBox[{{f, (, x, )}, x, a}, MinLimit2Arg]>=f(a). 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 :

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: