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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:

Limit from above:

The two-sided limit does not exist:

Scope  (5)

One-sided limits at piecewise discontinuities:

The default direction is two sided:

One-sided max limits:

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

Visualize the function:

Limits at a simple pole:

Limits at a branch cut:

Compute the bivariate limit approach from different quadrants:

Approaching the origin from the first quadrant:

Equivalently:

Approaching the origin from the second quadrant:

Approaching the origin from the left half-plane:

Approaching the origin from the bottom half-plane:

Visualize the function:

Properties & Relations  (4)

The settings "FromBelow" and 1 are equivalent:

The settings "FromAbove" and -1 are equivalent:

The settings "TwoSided" and Reals are equivalent:

Direction places conditions on the limit variable:

Assumptions applies to parameters in the limit expression:

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_2025_direction, author="Wolfram Research", title="{Direction}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/Direction.html}", note=[Accessed: 15-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_direction, organization={Wolfram Research}, title={Direction}, year={2017}, url={https://reference.wolfram.com/language/ref/Direction.html}, note=[Accessed: 15-August-2025]}