Limit

Limit[f[x],xx*]

gives the limit xx*f(x).

Limit[f[x1,…,xn],{x1,…,xn}]

gives the nested limit ⋯ f (x1,…,xn).

Limit[f[x1,…,xn],{x1,…,xn}{,…,}]

gives the multivariate limit f (x1,…,xn).

Details and Options

  • Limit is also known as function limit, directed limit, iterated limit, nested limit and multivariate limit.
  • Limit computes the limiting value f* of a function f as its variables x or xi get arbitrarily close to their limiting point x* or .
  • By using the character , entered as lim or \[Limit], with underscripts or subscripts, limits can be entered as follows:
  • flimit in the default direction
    flimit from above
    flimit from below
    flimit in the complex plane
    …fLimit[f,{x1,…,xn}]
  • For a finite limit point x* and {,…,} and finite limit value f*:
  • Limit[f[x],xx*]f*for every there is such that 0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<delta(epsilon,x^*) implies TemplateBox[{{{f, (, x, )}, -, {f, ^, *}}}, Abs]<epsilon
    Limit[f[x1,…,xn],{x1,…,xn}{,…,}]f*for every there is such that 0<TemplateBox[{{{, {{{x, _, 1}, -, {x, _, {(, 1, )}, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, {(, n, )}, ^, *}}}, }}}, Norm]<delta(epsilon,x^*) implies TemplateBox[{{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}, -, {f, ^, *}}}, Abs]<epsilon
  • For an infinite limit point and finite limit value f*:
  • Limit[f[x],x∞]f*for every there is such that implies TemplateBox[{{{f, (, x, )}, -, {f, ^, *}}}, Abs]<epsilon
    Limit[f[x1,…,xn],{x1,…,xn}{∞,…,∞}]f*for every there is such that implies TemplateBox[{{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}, -, {f, ^, *}}}, Abs]<epsilon
  • Limit returns Indeterminate when it can prove the limit does not exist. MinLimit and MaxLimit can frequently be used to compute the minimum and maximum limit of a function if its limit does not exist.
  • Limit returns unevaluated or an Interval when no limit can be found. If an Interval is returned, there are no guarantees that this is the smallest possible interval.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    DirectionRealsdirections to approach the limit point
    GenerateConditionsAutomaticwhether to generate conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"aspects of performance 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

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Basic Examples  (3)

Limit at a point of discontinuity:

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Limit at infinity:

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Limit from above:

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Limit from below:

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The two-sided limit does not exist:

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Scope  (35)

Options  (13)

Applications  (23)

Properties & Relations  (13)

Possible Issues  (1)

Interactive Examples  (1)

Neat Examples  (2)

See Also

DiscreteLimit  Series  Residue  MaxLimit  MinLimit  Derivative  Assumptions  DiracDelta  PrincipalValue

Tutorials

Introduced in 1988
(1.0)
| Updated in 2017
(11.2)