# 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 \$Assumptions assumptions on parameters GenerateConditions Automatic whether to generate conditions on parameters Method Automatic method to use PerformanceGoal "Quality" aspects of performance to optimize
• 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, 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:

## 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 :

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 , 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 :

Similarly, is said to be "big-theta of ", written if and :

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 :

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 :

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 :

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:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_discreteminlimit, author="Wolfram Research", title="{DiscreteMinLimit}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteMinLimit.html}", note=[Accessed: 19-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_discreteminlimit, organization={Wolfram Research}, title={DiscreteMinLimit}, year={2017}, url={https://reference.wolfram.com/language/ref/DiscreteMinLimit.html}, note=[Accessed: 19-July-2024 ]}