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Direction

is an option for Limit and related functions that specifies the direction in which the limit is taken.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
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History
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Direction

Direction

is an option for Limit and related functions that specifies the direction in which the limit is taken.

Details

  • 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*.

Examples

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Basic Examples  (1)

Limit from below:

Wolfram Language code: Limit[1 / x, x -> 0, Direction -> "FromBelow"]

Limit from above:

Wolfram Language code: Limit[1 / x, x -> 0, Direction -> "FromAbove"]

The two-sided limit does not exist:

Wolfram Language code: Limit[1 / x, x -> 0, Direction -> "TwoSided"]
Wolfram Language code: Plot[1 / x, {x, -1, 1}]

Scope  (5)

One-sided limits at piecewise discontinuities:

Wolfram Language code: Limit[RealAbs[x] / x, x -> 0, Direction -> "FromAbove"]
Wolfram Language code: Limit[RealAbs[x] / x, x -> 0, Direction -> "FromBelow"]

The default direction is two sided:

Wolfram Language code: Limit[RealAbs[x] / x, x -> 0]
Wolfram Language code: Plot[RealAbs[x] / x, {x, -1, 1}]

One-sided max limits:

Wolfram Language code: MaxLimit[UnitStep[x]Sin[(1/x)] + (1/2), x -> 0, Direction -> "FromAbove"]
Wolfram Language code: MaxLimit[UnitStep[x]Sin[(1/x)] + (1/2), x -> 0, Direction -> "FromBelow"]

The default direction Reals gives the larger of the two one-sided max limits:

Wolfram Language code: MaxLimit[UnitStep[x]Sin[(1/x)] + (1/2), x -> 0]
Wolfram Language code: MaxLimit[UnitStep[x]Sin[(1/x)] + (1/2), x -> 0, Direction -> Reals]

Visualize the function:

Wolfram Language code: Plot[UnitStep[x]Sin[(1/x)] + (1/2), {x, -.1, .1}]

Limits at a simple pole:

Wolfram Language code: Limit[Tan[x], x -> π / 2, Direction -> "FromBelow"]
Wolfram Language code: Limit[Tan[x], x -> π / 2, Direction -> "FromAbove"]
Wolfram Language code: Limit[Tan[x], x -> π / 2, Direction -> ℝ]
Wolfram Language code: Limit[Tan[x], x -> π / 2, Direction -> ℂ]

Limits at a branch cut:

Wolfram Language code: Limit[Sqrt[x], x -> -1, Direction -> Reals]
Wolfram Language code: Limit[Sqrt[x], x -> -1, Direction -> -I]
Wolfram Language code: Limit[Sqrt[x], x -> -1, Direction -> +I]
Wolfram Language code: Limit[Sqrt[x], x -> -1, Direction -> Complexes]

Compute the bivariate limit approach from different quadrants:

Wolfram Language code: f[x_, y_] := (x y/x + y)(2 + RealSign[ x y] + RealSign[y])

Approaching the origin from the first quadrant:

Wolfram Language code: Limit[f[x, y], {x, y} -> {0, 0}, Direction -> "FromAbove"]

Equivalently:

Wolfram Language code: Limit[f[x, y], {x, y} -> {0, 0}, Direction -> {"FromAbove", "FromAbove"}]

Approaching the origin from the second quadrant:

Wolfram Language code: Limit[f[x, y], {x, y} -> {0, 0}, Direction -> {"FromBelow", "FromAbove"}]

Approaching the origin from the left half-plane:

Wolfram Language code: Limit[f[x, y], {x, y} -> {0, 0}, Direction -> {"FromBelow", Reals}]

Approaching the origin from the bottom half-plane:

Wolfram Language code: Limit[f[x, y], {x, y} -> {0, 0}, Direction -> {Reals, "FromBelow"}]

Visualize the function:

Wolfram Language code: Row@{Plot3D[f[x, y], {x, y}∈Disk[], Exclusions -> {{x == -y , x <= 0}}, ClippingStyle -> None, ImageSize -> 200], DensityPlot[f[x, y], {x, -1, 1}, {y, -1, 1}, Exclusions -> {{x == -y , x <= 0}}, ClippingStyle -> Automatic, ImageSize -> Small]}

Properties & Relations  (4)

The settings "FromBelow" and 1 are equivalent:

Wolfram Language code: MinLimit[Cos[x] / x, x -> 0, Direction -> "FromBelow"]
Wolfram Language code: MinLimit[Cos[x] / x, x -> 0, Direction -> 1]

The settings "FromAbove" and -1 are equivalent:

Wolfram Language code: MinLimit[Cos[x] / x, x -> 0, Direction -> "FromAbove"]
Wolfram Language code: MinLimit[Cos[x] / x, x -> 0, Direction -> -1]

The settings "TwoSided" and Reals are equivalent:

Wolfram Language code: MinLimit[Cos[x] / x, x -> 0, Direction -> "TwoSided"]
Wolfram Language code: MinLimit[Cos[x] / x, x -> 0, Direction -> ℝ]

Direction places conditions on the limit variable:

Wolfram Language code: Limit[Tan[x], x -> π / 2 + n π, Direction -> 1, Assumptions -> n∈Integers]

Assumptions applies to parameters in the limit expression:

Wolfram Language code: Limit[Tan[x], x -> π / 2 + n π, Direction -> -1, Assumptions -> n∈Integers]
Wolfram Language code: Plot[Tan[x], {x, -2π, 2π}, ExclusionsStyle -> {{Dashed, Red}}]
Wolfram Research (1991), Direction, Wolfram Language function, https://reference.wolfram.com/language/ref/Direction.html (updated 2017).

Text

Wolfram Research (1991), Direction, Wolfram Language function, https://reference.wolfram.com/language/ref/Direction.html (updated 2017).

CMS

Wolfram Language. 1991. "Direction." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/Direction.html.

APA

Wolfram Language. (1991). Direction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Direction.html

BibTeX

@misc{reference.wolfram_2026_direction, author="Wolfram Research", title="{Direction}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/Direction.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_direction, organization={Wolfram Research}, title={Direction}, year={2017}, url={https://reference.wolfram.com/language/ref/Direction.html}, note=[Accessed: 12-June-2026]}