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:

Introduced in 1991
 (2.0)
 |
Updated in 2017
 (11.2)