DiscreteLimit

DiscreteLimit[f,k∞]

gives the limit k∞f(k) for the sequence f as k tends to infinity over the integers.

DiscreteLimit[f,{k1,…,kn}]

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

DiscreteLimit[f,{k1,…,kn}{,…,}]

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

Details and Options

  • DiscreteLimit is also known as discrete limit or limit over the integers.
  • DiscreteLimit computes the limiting value of a sequence f as its variables k or ki get arbitrarily large.
  • DiscreteLimit[f,k∞] can be entered as f. A template can be entered as dlim, and moves the cursor from the underscript to the body.
  • DiscreteLimit[f,{k1,…,kn}{,…,}] can be entered as …f.
  • The possible limit points are ±∞.
  • For a finite limit value f*:
  • DiscreteLimit[f,k∞]f*for every there is a such that implies TemplateBox[{{{f, (, k, )}, -, {f, ^, *}}}, Abs]<epsilon
    DiscreteLimit[f,{k1,…,kn}{∞,…,∞}]f*for every there is a such that implies TemplateBox[{{{f, (, {{k, _, 1}, ,, ..., ,, {k, _, n}}, )}, -, {f, ^, *}}}, Abs]<epsilon
  • DiscreteLimit[f[k],k-∞] is equivalent to DiscreteLimit[f[-l],l∞] etc.
  • DiscreteLimit returns Indeterminate when it can prove that the limit does not exist, and returns unevaluated when no limit can be found.
  • The following options can be given:
  • Assumptions $Assumptionsassumptions on parameters
    GenerateConditions Automaticwhether to generate conditions on parameters
    Method Automaticmethod 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, DiscreteLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (4)

Limit of a sequence:

Plot the sequence and its limit:

Limit of a multivariate sequence:

Plot the sequence and its limit:

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

TraditionalForm typesetting:

Scope  (37)

Basic Uses  (4)

Compute the limit of a sequence when n approaches Infinity:

Compute the 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  (7)

Find the limit of a rational sequence:

Geometric sequence:

Exponential sequences:

Trigonometric sequence:

Inverse trigonometric sequence:

Logarithmic sequence:

Find the limit of ArcTan[Log[n]]:

Integer Function Sequences  (5)

Compute the limit of a binomial sequence:

Limits of sequences involving FactorialPower:

Limits of sequences involving Factorial:

Compute limits involving Fibonacci and LucasL:

Limit involving Pochhammer:

Alternating Sequences  (3)

Convergent alternating sequence:

Divergent alternating sequence:

Oscillatory alternating sequence:

Periodic Sequences  (3)

Limits involving periodic sequences:

Eventually periodic sequence:

Densely aperiodic sequences:

Piecewise Function Sequences  (3)

A convergent piecewise sequence:

A divergent piecewise sequence:

Piecewise sequence with periodic conditions:

Limit involving Floor:

Number Theoretic Function Sequences  (4)

Compute limits involving Prime:

Prime is of order :

Limits involving PrimePi:

PrimePi is of order :

Limits involving PartitionsP and PartitionsQ:

Limits involving other number theoretic sequences:

Nested and Multivariate Sequences  (2)

Compute a nested sequence limit:

Plot the sequence and its limit:

Multivariate sequence limits:

Formal Sequences  (6)

Compute limits of sequences involving Inactive sums:

Nested limit of an Inactive sum:

Obtain the same result in two steps using an interchange of DiscreteLimit and Sum:

Limits of sequences involving Inactive products:

Nested limit of an Inactive product:

Obtain the same result in two steps using an interchange of DiscreteLimit and Product:

Limits of sequences involving Inactive continued fractions:

Nested limit of an Inactive continued fraction:

Obtain the same result in two steps using DiscreteLimit and ContinuedFractionK:

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 y>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 limit of a sequence using the default method:

Obtain the same answer using a call to Limit:

The given sequence is not periodic, hence the method for periodic sequences fails:

PerformanceGoal  (1)

DiscreteLimit computes limits involving sequences of arbitrarily large periods:

Use PerformanceGoal to avoid potentially expensive computations in such cases:

The Method option overrides PerformanceGoal:

Applications  (35)

Geometric Limits  (3)

The perimeter of a regular polygon of radius r and n sides:

In the limit n->∞, the perimeter approaches the circumference of a circle of radius r:

The area of a regular polygon of radius r and n sides:

In the limit n->∞, this approaches the area of a circle of radius r:

Visualize the inscribed polygon and the approximate perimeter and area as n increases:

Consider covering a ball of radius r by 2n cylinders as shown in the figure:

The volume of the cylinders is:

Taking the DiscreteLimit as n->Infinity gives the volume of the ball:

Compare with a direct computation:

Consider the following function and a set of rectangles defined by its plot:

For n5 on the interval [0,2], the rectangles are the following:

The area of these rectangles defines a Riemann sum that approximates the area under the curve:

Use DiscreteLimit to obtain the exact answer:

Obtain the same area directly using Integrate:

Visualize the process for this function as well as three others:

Sums and Products  (6)

Compute an infinite sum as the limit of a finite sum:

Obtain the same answer using Sum:

The following sequence defines a convergent series:

Find the value of the series:

Compute the result directly using Sum:

Prove that an infinite series is divergent, starting with the sum of a finite number of terms:

The series diverges, since the limit of the finite sums does not exist:

Confirm the divergence using SumConvergence and Sum:

Obtain the Abel sum of the series using Regularization:

Compute a doubly infinite sum as a nested limit of a finite sum:

Obtain the same answer directly using Sum:

Compute an infinite product as a limit of a finite product:

Obtain the same answer using Product:

Construct a rotation matrix as a limit of repeated infinitesimal transformations:

Compare with a direction construction:

Series Convergence  (4)

Use the ratio test to verify convergence of a series whose general term is given by:

Compute the DiscreteRatio for this series:

The series converges, since the limit of the ratio is less than 1:

Verify the result using SumConvergence:

Use the root test to verify convergence of a series whose general term is given by:

The series converges, since the limit of the n^(th) root is less than 1:

Verify the result using SumConvergence:

Use the Raabe test to verify convergence of a series whose general term is given by:

Raabe's test applies because the ratio test is inconclusive:

The series converges, since the following limit is greater than 1:

Verify the result using SumConvergence:

Use the divergence test to verify divergence of a series whose general term is given by:

The series diverges, since the limit of the general term is not 0:

Verify the result using SumConvergence:

Classical Definition  (3)

Show that the following sequence converges to 0, and verify the classical definition with ϵ=1/7:

Compute the limit:

Set the value of :

Use Reduce to show that the definition is satisfied for all n>=12:

Verify the result using DiscretePlot:

Show that the following sequence diverges to Infinity, and verify the classical definition with M=35:

Compute the limit:

Set the value of M:

Use Reduce to show that the definition is satisfied for all n >= 10:

Verify the result using DiscretePlot:

Determine the convergence of the harmonic series , whose terms are given by:

Standard tests such as the ratio test are inconclusive:

Define an auxiliary series as follows:

The terms of consist of runs of length of :

Notice that :

Also, the sum of each run is , so the sum of the first terms is :

The partial sums of are called the harmonic numbers TemplateBox[{n}, HarmonicNumber]:

For any positive integer , TemplateBox[{{2, ^, {(, {{2, , M}, -, 2}, )}}}, HarmonicNumber]>=sum_(n=1)^(2^(2 M-2))a(n)=M, so TemplateBox[{n}, HarmonicNumber] eventually exceeds and diverges to :

This means the sum of does not converge:

The divergence is slow, however, requiring more than terms just to get over :

Recursive Sequences  (3)

Compute the limit of a nonlinear recursive sequence that is specified using RSolveValue:

Compute the limit of a trigonometric recursive sequence that is specified using RSolveValue:

Compute the value of :

Mathematical Constants  (5)

Compute as the limit of a sequence:

Compute as the limit of a Sum:

Compute as the limit of a sequence:

Compute EulerGamma using the limit of a sequence:

Compute the golden ratio using a sequence involving Fibonacci:

Mathematical Functions  (2)

Represent as the limit of a sequence with symbolic entries:

Represent Log[x] as the limit of a sequence:

Stolz–Cesàro Theorem  (2)

The Stolz–Cesàro theorem is a discrete version of L'Hôpital's rule, and can be used to compute the limits for ratios of sequences, under suitable conditions. The theorem states that:

Verify the Stolz–Cesàro theorem for the sequences defined by:

Compute the limit for the ratio of differences:

Obtain the same result directly using DiscreteLimit:

Plot the sequence and the limit:

Verify the Stolz–Cesàro theorem for the sequences defined by:

Compute the limit for the ratio of differences:

Obtain the same result directly using DiscreteLimit:

Plot the sequence and the limit:

Computational Complexity  (3)

An algorithm runtime function is said to be "little-o of ", written , if :

Similarly, is said to be "little-omega of ", written , if :

If , then :

It is possible for two functions to share neither relationship:

Moreover, neither relationship even holds between a function and itself:

Hence, and define partial orders on the space of algorithm runtimes:

if the algorithm associated to is much faster than the one associated to for large inputs:

denotes the opposite relationship:

Note that the two lists are not exactly reversed, because and are incomparable:

An algorithm runtime function is said to be "big-theta of ", written , if the following holds:

Consider an algorithm that takes time —a polynomial of degree —to run:

The ratio of this function to the monomial goes to the leading coefficient at infinity:

Since the limit of the sequence exists, its max and min limits must both equal this value:

For an algorithmic runtime, must be a positive finite number, so every polynomial algorithm is :

Hence, only the leading term in the polynomial is important in determining the runtime for large inputs:

Check the asymptotic complexity of the fast Fourier transform:

Compute the asymptotic complexity:

Uniform Convergence  (2)

At every point , the following sequence of functions converges to zero:

The greatest magnitude of each is achieved at :

Thus, for any , implies that TemplateBox[{{{f, _, n}, (, x, )}}, RealAbs]<epsilon for all and the convergence is uniform:

As a consequence, the limit of the integrals equals the integral of the limit:

At every point , the following sequence of functions converges to zero:

However, the maximum value of , at the point , diverges as :

This shows that the convergence of the sequence of functions is not uniform:

As a consequence, the limit of the integrals does not equal the integral of the limit:

Miscellaneous Applications  (2)

Compute the inverse Laplace transform of using Post's inversion formula:

The inverse Laplace transform of this function is 1:

Obtain the same result using InverseLaplaceTransform:

Create a table of basic inverse Laplace transforms using Post's inversion formula:

The limit of the probability distribution for a sequence of random variables, if it exists, is called an asymptotic distribution. Obtain the Poisson distribution as an asymptotic distribution for a sequence of binomial distributions in which the mean value λ, the product of the probability and number of trials, is held constant:

Compute the limit of this sequence as the number of trials n->∞:

Verify that this is the PDF for PoissonDistribution:

Plot the distributions for λ=8 and various values of n. Notice that the PDF is zero for all k>n:

Properties & Relations  (15)

Multiplicative constants can be moved outside a limit:

If f and g have finite limits, DiscreteLimit is distributive over a sum:

If f and g have finite limits, DiscreteLimit is distributive over a product:

Powers can be moved outside a limit:

Function composition and sequence limit operations can be interchanged for continuous functions:

This need not hold for discontinuous functions:

The "squeezing" or "sandwich" theorem:

This function is bounded by on the positive integers:

The limit of the bounding functions is zero, which proves the original limit was zero:

The Stolz–Cesàro rule can be used to find the limit of the ratio of two sequences:

Directly solving the limit leads to an indeterminate form of type :

The Stolz–Cesàro rule is applied to correctly compute the limit:

If Limit exists, then so does DiscreteLimit, and they have the same value:

The converse need not hold:

If DiscreteLimit exists, then so does DiscreteMaxLimit, and they have the same value:

If DiscreteLimit exists, then so does DiscreteMinLimit, and they have the same value:

The limit of a difference satisfies TemplateBox[{{{f, (, {n, +, 1}, )}, -, {f, (, n, )}}, n, infty}, DiscreteLimit]=TemplateBox[{{{(, {f, (, n, )}, )}, /, n}, n, infty}, DiscreteLimit]:

The limit of a ratio satisfies TemplateBox[{{{(, {f, (, {n, +, 1}, )}, )}, /, {(, {f, (, n, )}, )}}, n, infty}, DiscreteLimit]=TemplateBox[{{{f, (, n, )}, ^, {(, {1, /, n}, )}}, n, infty}, DiscreteLimit]:

Compute the limit of a sequence using a finite sum:

Compute the limit of a sequence using a finite product:

The limit of a sequence is related to its ZTransform via the final value theorem:

Verify the final value theorem:

Neat Examples  (1)

Visualize a set of sequence limits:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_discretelimit, author="Wolfram Research", title="{DiscreteLimit}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteLimit.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_discretelimit, organization={Wolfram Research}, title={DiscreteLimit}, year={2017}, url={https://reference.wolfram.com/language/ref/DiscreteLimit.html}, note=[Accessed: 21-December-2024 ]}