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AsymptoticEqual
✖
AsymptoticEqual

✖
AsymptoticEqual[f,g,xx*]

gives conditions for or as xx*.

✖
AsymptoticEqual[f,g,{x1,…,xn}{,…,}]

gives conditions for or as {x1,…,xn}{,…,}.

Details and Options

  • Asymptotic equal is also expressed as f is big-theta of g, f is bounded by g, f is of order g, and f grows as g. The point x* is often assumed from context.
  • Asymptotic equal is an equivalence relation and means c_1 TemplateBox[{{g, (, x, )}}, Abs]<=TemplateBox[{{f, (, x, )}}, Abs]<=c_2 TemplateBox[{{g, (, x, )}}, Abs] when x is near x* for some constants and . It is a coarser asymptotic equivalence relation than AsymptoticEquivalent.
  • Typical uses include expressing simple bounds for functions and sequences near some point. It is frequently used for asymptotic solutions to equations and to give simple lower bounds for computational complexity.
  • For a finite limit point x* and {,…,}, the result is:
  • AsymptoticEqual[f[x],g[x],xx*]there exist , and such that 0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<delta(c_1,c_2,x^*) implies c_1 TemplateBox[{{g, (, x, )}}, Abs]<=TemplateBox[{{f, (, x, )}}, Abs]<=c_2 TemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticEqual[f[x1,…,xn],g[x1,…,xn],{x1,…,xn}{,…,}]there exist , and such that 0<TemplateBox[{{{, {{{x, _, 1}, -, {x, _, {(, 1, )}, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, {(, n, )}, ^, *}}}, }}}, Norm]<delta(epsilon,x^*) implies c_1 TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=TemplateBox[{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=c_2 TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]
  • For an infinite limit point, the result is:
  • AsymptoticEqual[f[x],g[x],x∞]there exist , and such that implies c_1 TemplateBox[{{g, (, x, )}}, Abs]<=TemplateBox[{{f, (, x, )}}, Abs]<=c_2 TemplateBox[{{g, (, x, )}}, Abs]
    AsymptoticEqual[f[x1,…,xn],g[x1,…,xn],{x1,…,xn}{∞,…,∞}]there exist , and such that implies c_1 TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=TemplateBox[{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]<=c_2 TemplateBox[{{g, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}}, Abs]
  • AsymptoticEqual[f[x],g[x],xx*] exists if and only if MinLimit[Abs[f[x]/g[x]],xx*]>0 and MaxLimit[Abs[f[x]/g[x]],xx*]<∞ when g[x] does not have an infinite set of zeros near x*.
  • The following options can be given:
  • Assumptions $Assumptionsassumptions on parameters
    Direction Realsdirection to approach the limit point
    GenerateConditions Automaticgenerate conditions for parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"what to optimize
  • Possible settings for Direction include:
  • Reals or "TwoSided"from both real directions
    "FromAbove" or -1from above or larger values
    "FromBelow" or +1from below or smaller values
    Complexesfrom all complex directions
    Exp[ θ]in the direction
    {dir1,…,dirn}use direction diri for variable xi independently
  • DirectionExp[ θ] at x* indicates the direction tangent of a curve approaching the limit point x*.
  • 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, Limit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Verify that as :

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-fd2m9a
Out[1]=1

The former can be "sandwiched" between two constant multiples of the latter:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-ekzxa6
Out[3]=3

Verify that as :

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-pwvtd
Out[1]=1

The former can be "sandwiched" between two constant multiples of the latter:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-bah9
Out[2]=2

Scope  (9)Survey of the scope of standard use cases

Compare functions that are not strictly positive:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-jc83n1
Out[1]=1

Show that diverges at the same rate as at the origin:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-nnsqde
Out[1]=1

Answers may be Boolean expressions rather than explicit True or False:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-sxh24
Out[1]=1

When comparing functions with parameters, conditions for the result may be generated:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-w0qbz9
Out[1]=1

By default, a two-sided comparison of the functions is made:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-gclur
Out[1]=1

When comparing larger values of , vanishes at the same rate as x^2(1-cos(x)) TemplateBox[{{x}}, UnitStepSeq]:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-szc5cz
Out[2]=2

The relationship fails for smaller values of :

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-d90890
Out[3]=3

Visualize the ratio of the two functions, showing it approaches a nonzero limit from above:

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-2mykrk
Out[4]=4

Functions like Sqrt may have the same relationship in both real directions along the negative reals:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-pgmwgz
Out[1]=1

If approached from above in the complex plane, the same relationship is observed:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-eh3ic2
Out[2]=2

However, approaching from below in the complex plane produces a different result:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-7oj7pv
Out[3]=3

This is due to a branch cut where the imaginary part of Sqrt reverses sign as the axis is crossed:

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-xj6jic

Hence, the relationship does not hold in the complex plane in general:

In[5]:=5
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-9gewpp
Out[5]=5

Visualize the relative sizes of the functions when approached from the four real and imaginary directions:

In[6]:=6
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-ejoqih
Out[6]=6

Compare multivariate functions:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-485nrb
Out[1]=1

Visualize the norms of the two functions:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-w2pict
Out[2]=2

Compare multivariate functions at infinity:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-2pjlet
Out[1]=1

Use parameters when comparing multivariate functions:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-jkkb89
Out[1]=1

Options  (9)Common values & functionality for each option

Assumptions  (1)

Specify conditions on parameters using Assumptions:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-bsi62b
Out[1]=1

Different assumptions can produce different results:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-9nq9mt
Out[2]=2

Direction  (5)

Equality from below:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-wh362x
Out[1]=1

Equivalently:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-47rpfm
Out[2]=2

Equality from above:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-ieye3l
Out[3]=3

Equivalently:

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-7b33iw
Out[4]=4

Equality at piecewise discontinuities:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-dtc1xq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-1s90kg
Out[2]=2

Since it fails in one direction, the two-sided result is false as well:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-ce2ne8
Out[3]=3

Visualize the two functions and their ratio:

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-fpzs
Out[4]=4

Equality at a pole is independent of the direction of approach:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-o19pbz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-4oak0w
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-6gbhpd
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-7wy1b3
Out[4]=4

Equality at a branch cut:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-bsn6zf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-2fd8pf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-7508fb
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-iio0jj
Out[4]=4

Compute equality, approaching from different quadrants:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-xstf5t
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-n5ghf

Approaching the origin from the third quadrant:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-5ei7y4
Out[3]=3

Equivalently:

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-0n4vor
Out[4]=4

Approaching the origin from the second quadrant:

In[5]:=5
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-772jnn
Out[5]=5

Approaching the origin from the right half-plane:

In[6]:=6
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-pa15d2
Out[6]=6

Approaching the origin from the bottom half-plane:

In[7]:=7
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-46gbl5
Out[7]=7

Visualize the ratio of the functions:

In[8]:=8
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-ry52i7
Out[8]=8

GenerateConditions  (3)

Return a result without stating conditions:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-6rd3fa
Out[1]=1

This result is only valid if n>0:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-uhf4he
Out[1]=1
In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-1er301
Out[3]=3

Return unevaluated if the results depend on the value of parameters:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-2lepxp
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-1s089w
Out[2]=2

By default, conditions are not generated if only special values invalidate the result:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-tdcquw
Out[1]=1

With GenerateConditions->True, even these non-generic conditions are reported:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-291b1m
Out[2]=2

Applications  (10)Sample problems that can be solved with this function

Basic Applications  (5)

Show that two monomials equivalent at infinity have the same power:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-xlwmnq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-3gga8g
Out[2]=2

Use this to show that two polynomials that are equivalent are of the same order:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-v7sj5o
Out[3]=3

Visualize the ratios of three pairs of asymptotically equivalent polynomials:

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-0o3ud1
Out[4]=4

Show that two monomials in equivalent at have the same power:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-49j3mw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-hognh6
Out[2]=2

Use this to show that two polynomials in that are equivalent have the same leading monomial:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-33ul3n
Out[3]=3

Visualize the ratios of three pairs of asymptotically equivalent polynomials in :

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-f412xf
Out[4]=4

Show that at :

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-kmoq7r
Out[1]=1

Visualize the functions:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-yt01k0
Out[2]=2

Even though the absolute value of their ratio wiggles incessantly as , it is bounded away from :

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-l8frvd
Out[3]=3

Show that :

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-ddqabt
Out[1]=1

Even though the absolute value of their ratio wiggles incessantly as , it is bounded away from :

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-x99r0n
Out[2]=2

Show that at :

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-8oriwy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-w7jyr7
Out[2]=2

Computational Complexity  (3)

In a bubble sort, adjoining neighbors are compared and swapped if they are out of order. After one pass of n-1 comparisons, the largest element is at the end. The process is then repeated on remaining n-1 elements, and so forth, until only two elements at the very beginning remain. If comparison and swap takes c steps, the total number of steps for the sort is as follows:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-g63m1r
Out[1]=1

Show that , and thus the algorithm has quadratic run time:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-8x1qob
Out[2]=2

Visualize the ratios for the two functions for different values of c:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-lwgbuf
Out[3]=3

In a merge sort, the list of elements is split in two, each half is sorted, and then the two halves are combined. Thus, the time T[n] to do the sort will be the sum of some constant time b to compute the middle, 2T[n/2] to sort each half, and some multiple a n of the number of elements to combine the two halves:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-hgn32u
Out[1]=1

Solve the recurrence equation to find the time t to sort n elements:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-nn6dxo
Out[2]=2

Show that , and thus the algorithm has run time:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-u1dsrr
Out[3]=3

Strassen's algorithm was the first subcubic matrix multiplication algorithm found. It consists of four steps: splitting each of the two matrices into 4 equal-sized submatrices, forming 14 particular linear combinations out of the 8 submatrices, multiplying 7 pairs of these and forming linear combinations of the 7 results. The time to do the multiplication will therefore be a constant time to split the matrix, for forming linear combinations in the second and fourth steps and for the third step:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-okd8ll
Out[1]=1

Solve the recurrence relation:

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-qbdv82
Out[2]=2

Though the result is complicated, it is asymptotically equal to :

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-xadpzc
Out[3]=3

Compare the rate of growth of the naive cubic algorithm and Strassen's algorithm:

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-o4nj4q
Out[4]=4

Convergence Testing  (2)

A sequence is said to be absolutely summable if sum_(n=1)^inftyTemplateBox[{{a, _, n}}, Abs]<infty. If a second sequence , the comparison test states that is absolutely summable if and only if is. Use the test to show that converges by comparing with the sum of :

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-c6ak1d
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-focct0
Out[2]=2

Compare with the answer given by SumConvergence:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-n3mdiq
Out[3]=3

Show that convergence by comparing with the sum of :

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-5fvg9h
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-732a8a
Out[5]=5

Show that divergence by comparing with the sum of :

In[6]:=6
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-tg0kcf
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-y36z5o
Out[7]=7

Compare with the answer given by SumConvergence:

In[8]:=8
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-6xnxna
Out[8]=8

The sequence , and thus is not absolutely summable:

In[9]:=9
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-sea9r4
Out[9]=9
In[10]:=10
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-6gu07k
Out[10]=10

A function is said to be absolutely integrable on if int_a^bTemplateBox[{{f, , {(, x, )}}}, Abs]dx<infty. If and are continuous on the open interval and at both and , the comparison test states that is absolutely integrable if and only if is. Use the test to show that is absolutely integrable on :

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-jqu98u
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-0mll1r
Out[2]=2

Since is integrable on , so is :

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-rvsvtz
Out[3]=3

That is not integrable over can be shown by comparing it with :

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-sak382
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-n1as9a
Out[5]=5

The result follows from the nonintegrability of on the unit interval:

In[6]:=6
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-cosev6
Out[6]=6

The function is not absolutely integrable on :

In[7]:=7
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-v7866i
Out[7]=7

Show that TemplateBox[{x}, LogIntegral]/x is not absolutely integrable either, by comparing it to :

In[8]:=8
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-2hi0mp
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-ti2r3w
Out[9]=9

Properties & Relations  (6)Properties of the function, and connections to other functions

AsymptoticEqual is an equivalence relation, meaning it is reflexive ():

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-djhovi
Out[1]=1

It is transitive ( and implies ):

In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-iflbs4
In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-jo9vs
Out[3]=3

And it is symmetric ( implies ):

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-2nf1s
Out[4]=4

AsymptoticEqual[f[x],g[x],xx0] iff MaxLimit[Abs[f[x]/g[x]],xx0]<∞ and MinLimit[Abs[f[x]/g[x]],xx0]>0:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-jjmtdb
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-bwsd3m
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-jqjzfw
In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-fdt3ec
Out[4]=4

AsymptoticEqual[f[x],g[x],xx0] if 0<Limit[Abs[f[x]/g[x]],xx0]<∞:

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-d20hmb
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-jy4cm4
Out[2]=2

However, the limit need not exist:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-5766nc
In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-79sson
Out[4]=4

iff and :

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-f0y2os
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-y915f
Out[2]=2

If or , then

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-z10kcc
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-jnddz8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-3jo0a6
Out[3]=3

The converse, however, is false:

In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-h1p3hx
In[5]:=5
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-441xqo
Out[5]=5

If then :

In[1]:=1
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-fhpchq
In[2]:=2
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-bgd4x
Out[2]=2

The converse is false, so AsymptoticEqual is coarser than AsymptoticEquivalent:

In[3]:=3
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-m24dj
In[4]:=4
✖
https://wolfram.com/xid/01lrfsocn65rcn8re-ckmxl0
Out[4]=4
Wolfram Research (2018), AsymptoticEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticEqual.html.
✖
Wolfram Research (2018), AsymptoticEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticEqual.html.

Text

Wolfram Research (2018), AsymptoticEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticEqual.html.

✖
Wolfram Research (2018), AsymptoticEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticEqual.html.

CMS

Wolfram Language. 2018. "AsymptoticEqual." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AsymptoticEqual.html.

✖
Wolfram Language. 2018. "AsymptoticEqual." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AsymptoticEqual.html.

APA

Wolfram Language. (2018). AsymptoticEqual. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AsymptoticEqual.html

✖
Wolfram Language. (2018). AsymptoticEqual. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AsymptoticEqual.html

BibTeX

@misc{reference.wolfram_2025_asymptoticequal, author="Wolfram Research", title="{AsymptoticEqual}", year="2018", howpublished="\url{https://reference.wolfram.com/language/ref/AsymptoticEqual.html}", note=[Accessed: 23-April-2025 ]}

✖
@misc{reference.wolfram_2025_asymptoticequal, author="Wolfram Research", title="{AsymptoticEqual}", year="2018", howpublished="\url{https://reference.wolfram.com/language/ref/AsymptoticEqual.html}", note=[Accessed: 23-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_asymptoticequal, organization={Wolfram Research}, title={AsymptoticEqual}, year={2018}, url={https://reference.wolfram.com/language/ref/AsymptoticEqual.html}, note=[Accessed: 23-April-2025 ]}

✖
@online{reference.wolfram_2025_asymptoticequal, organization={Wolfram Research}, title={AsymptoticEqual}, year={2018}, url={https://reference.wolfram.com/language/ref/AsymptoticEqual.html}, note=[Accessed: 23-April-2025 ]}