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Direction
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Direction

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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

open allclose all

Basic Examples  (1)Summary of the most common use cases

Limit from below:

In[1]:=1
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https://wolfram.com/xid/0ftvl3m-j1osz
Out[1]=1

Limit from above:

In[2]:=2
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https://wolfram.com/xid/0ftvl3m-cvqtb9
Out[2]=2

The two-sided limit does not exist:

In[3]:=3
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https://wolfram.com/xid/0ftvl3m-esipz2
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ftvl3m-4631j
Out[4]=4

Scope  (5)Survey of the scope of standard use cases

One-sided limits at piecewise discontinuities:

In[1]:=1
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https://wolfram.com/xid/0ftvl3m-ipudde
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ftvl3m-ft6e6b
Out[2]=2

The default direction is two sided:

In[3]:=3
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https://wolfram.com/xid/0ftvl3m-gywhkj
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ftvl3m-hdpygy
Out[4]=4

One-sided max limits:

In[1]:=1
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https://wolfram.com/xid/0ftvl3m-uvk2q0
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ftvl3m-2aqxhp
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0ftvl3m-t7wscu
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ftvl3m-orpzfs
Out[4]=4

Visualize the function:

In[5]:=5
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https://wolfram.com/xid/0ftvl3m-u901c5
Out[5]=5

Limits at a simple pole:

In[1]:=1
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https://wolfram.com/xid/0ftvl3m-o19pbz
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ftvl3m-fil06t
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ftvl3m-rskvm9
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ftvl3m-pwklx7
Out[4]=4

Limits at a branch cut:

In[1]:=1
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https://wolfram.com/xid/0ftvl3m-l9tz2o
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ftvl3m-bfqizs
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ftvl3m-bsn6zf
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ftvl3m-bznk1n
Out[4]=4

Compute the bivariate limit approach from different quadrants:

In[1]:=1
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https://wolfram.com/xid/0ftvl3m-7ig6o1

Approaching the origin from the first quadrant:

In[2]:=2
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https://wolfram.com/xid/0ftvl3m-5ei7y4
Out[2]=2

Equivalently:

In[3]:=3
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https://wolfram.com/xid/0ftvl3m-1qv0hq
Out[3]=3

Approaching the origin from the second quadrant:

In[4]:=4
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https://wolfram.com/xid/0ftvl3m-r5ddxu
Out[4]=4

Approaching the origin from the left half-plane:

In[5]:=5
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https://wolfram.com/xid/0ftvl3m-9ebl17
Out[5]=5

Approaching the origin from the bottom half-plane:

In[6]:=6
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https://wolfram.com/xid/0ftvl3m-2gbxu
Out[6]=6

Visualize the function:

In[7]:=7
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https://wolfram.com/xid/0ftvl3m-ig2inr
Out[7]=7

Properties & Relations  (4)Properties of the function, and connections to other functions

The settings "FromBelow" and 1 are equivalent:

In[1]:=1
✖
https://wolfram.com/xid/0ftvl3m-wh362x
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ftvl3m-5b4ga
Out[2]=2

The settings "FromAbove" and -1 are equivalent:

In[1]:=1
✖
https://wolfram.com/xid/0ftvl3m-kwk8vy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ftvl3m-jvdswb
Out[2]=2

The settings "TwoSided" and Reals are equivalent:

In[1]:=1
✖
https://wolfram.com/xid/0ftvl3m-i77nhq
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ftvl3m-yilv3b
Out[2]=2

Direction places conditions on the limit variable:

In[1]:=1
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https://wolfram.com/xid/0ftvl3m-ex8lz
Out[1]=1

Assumptions applies to parameters in the limit expression:

In[2]:=2
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https://wolfram.com/xid/0ftvl3m-dyj6y
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ftvl3m-bqlxa9
Out[3]=3
Wolfram Research (1991), Direction, Wolfram Language function, https://reference.wolfram.com/language/ref/Direction.html (updated 2017).
✖
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).

✖
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.

✖
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

✖
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: 06-April-2025 ]}

✖
@misc{reference.wolfram_2025_direction, author="Wolfram Research", title="{Direction}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/Direction.html}", note=[Accessed: 06-April-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: 06-April-2025 ]}

✖
@online{reference.wolfram_2025_direction, organization={Wolfram Research}, title={Direction}, year={2017}, url={https://reference.wolfram.com/language/ref/Direction.html}, note=[Accessed: 06-April-2025 ]}