DiscreteMinLimit

DiscreteMinLimit[f,k]

gives the min limit kf(k) of the sequence f as k tends to over the integers.

DiscreteMinLimit[f,{k1,,kn}]

gives the nested min limit f(k1,,kn) over the integers.

DiscreteMinLimit[f,{k1,,kn}{,,}]

gives the multivariate min limit f(k1,,kn) over the integers.

Details and Options

  • DiscreteMinLimit is also known as limit inferior, infimum limit, liminf, lower limit and inner limit.
  • DiscreteMinLimit computes the largest lower bound for the limit and is always defined for real-valued sequences. It is often used to give conditions of convergence and other asymptotic properties that do not rely on an actual limit to exist.
  • DiscreteMinLimit[f,k] can be entered as f. A template can be entered as dmlim, and moves the cursor from the underscript to the body.
  • DiscreteMinLimit[f,{k1,,kn}{,,}] can be entered asf.
  • The possible limit points are ±.
  • The min limit is defined as a limit of the min envelope sequence min[ω]:
  • DiscreteMinLimit[f,k]DiscreteLimit[min[ω],ω]
    DiscreteMinLimit[f,{k1,,kn}{,,}]DiscreteLimit[min[ω],ω]
  • DiscreteMinLimit[f[k],k-] is equivalent to DiscreteMinLimit[f[-l],l] etc.
  • The definition uses the min envelope min[ω]MinValue[{f[k],kωk},k] for univariate f[k] and min[ω]MinValue[{f[k1,,kn],k1ωknωki},{k1,,kn}] for multivariate f[k1,,kn]. The sequence min[ω] is monotone increasing as ω, so always has a limit, which may be ±.
  • The illustration shows min[k] and min[Min[k1,k2]] in blue.
  • DiscreteMinLimit returns unevaluated when the min limit cannot be found.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    GenerateConditionsAutomaticwhether to generate conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"aspects of performance to optimize
  • Possible settings for GenerateConditions include:
  • Automaticnon-generic conditions only
    Trueall conditions
    Falseno conditions
    Nonereturn unevaluated if conditions are needed
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, DiscreteMinLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (4)

Min limit of a sequence:

Min limit of a product:

Use dmlim to enter the template and to move from the underscript to the body:

TraditionalForm typesetting:

Scope  (22)

Basic Uses  (4)

Compute the min limit of a sequence when n approaches Infinity:

Compute the min limit of a sequence when n approaches -Infinity:

Compute a nested limit for a multivariate sequence:

Compute the limit of a list of sequences:

Elementary Function Sequences  (6)

Find the min limit of a rational-exponential sequence:

Convergent geometric sequence:

Oscillating geometric sequence:

Divergent oscillating geometric sequence:

Exponential sequence:

Power sequence:

Trigonometric sequences:

Inverse trigonometric sequence:

Logarithmic sequence:

Integer Function Sequences  (3)

Sequences involving Factorial:

Sequence involving FactorialPower:

Compute the limit of a sequence involving Fibonacci:

Periodic Sequences  (3)

Limits of periodic sequences:

Eventually periodic sequence:

Densely aperiodic sequences:

Piecewise Function Sequences  (2)

Piecewise sequence with a finite min limit:

Piecewise sequence with an infinite min limit:

Piecewise sequence with periodic conditions:

Number Theoretic Sequences  (2)

Limits involving LCM and GCD:

Sequence involving Prime:

Multivariate Sequences  (2)

Compute a nested min limit:

Plot the sequence and its limit:

Multivariate min limits:

Options  (6)

Assumptions  (1)

Specify assumptions on a parameter:

Different assumptions can produce different results:

GenerateConditions  (3)

Return a result without stating conditions:

This result is only valid if x>1:

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:

Method  (1)

Compute the min limit of a periodic sequence using the default method:

Obtain the same answer using the method for periodic sequences:

The limit of the sequence is undefined, since it oscillates between 0 and 1:

PerformanceGoal  (1)

DiscreteMinLimit computes limits involving sequences of arbitrarily large periods:

Use PerformanceGoal to avoid potentially expensive computations in such cases:

The Method option overrides PerformanceGoal:

Applications  (3)

Compute the asymptotic minimum of a sequence:

Plot the sequence and the asymptotic minimum:

Verify that the following sequence does not have a limit:

Show that DiscreteMaxLimit and DiscreteMinLimit are not equal:

Confirm that the limit does not exist by using DiscreteLimit:

An algorithm runtime function is said to be "big-omega of ", written , if _(n->_(TemplateBox[{}, Integers])infty) (f(n))/(g(n))>0:

Similarly, is said to be "big-theta of ", written if _(n->_(TemplateBox[{}, Integers])infty) (f(n))/(g(n))<infty and _(n->_(TemplateBox[{}, Integers])infty)(f(n))/(g(n))>0:

The statement is always true:

If and , then :

It is possible for two functions to share neither relationship:

Hence, defines a reflexive partial order on the space of algorithm runtimes similar to :

If and , then , which implies that is an equivalence relation:

Properties & Relations  (11)

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

The corresponding limit may not exist:

If and have finite min limits, then TemplateBox[{{(, {f, +, g}, )}, x, a}, DiscreteMinLimit]>=TemplateBox[{f, x, a}, DiscreteMinLimit]+TemplateBox[{g, x, a}, DiscreteMinLimit]:

In this case, there is strict inequality:

Positive multiplicative constants can be moved outside a limit:

For a real-valued sequence, if DiscreteLimit exists, DiscreteMinLimit has the same value:

If has a finite limit, then TemplateBox[{{(, {f, +, g}, )}, x, a}, DiscreteMinLimit]=TemplateBox[{f, x, a}, DiscreteMinLimit]+TemplateBox[{g, x, a}, DiscreteMinLimit]:

DiscreteMinLimit is less than or equal to DiscreteMaxLimit:

If DiscreteMinLimit equals DiscreteMaxLimit, the limit exists and equals their common value:

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

DiscreteMinLimit can be computed as -DiscreteMaxLimit[-f,]:

If , then TemplateBox[{{g, (, n, )}, x, a}, MinLimit2Arg]>=TemplateBox[{{f, (, n, )}, x, a}, MaxLimit2Arg]>=TemplateBox[{{f,  , {(, n, )}}, x, a}, MinLimit2Arg]:

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

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

MinLimit is always less than or equal to DiscreteMinLimit:

Possible Issues  (1)

DiscreteMinLimit is only defined for real-valued sequences:

Neat Examples  (1)

Visualize a set of sequence min limits:

Introduced in 2017
 (11.2)