# 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:
• f limit in the default direction f limit from above f limit from below f limit in the complex plane … f Limit[f,{x1 ,…,xn }]
• For a finite limit point x* and { ,, } and finite limit value f*:
•  Limit[f[x],xx*]f* for every there is such that implies Limit[f[x1,…,xn],{x1,…,xn}{ ,…, }]f* for every there is such that implies • • For an infinite limit point and finite limit value f*:
•  Limit[f[x],x∞]f* for every there is such that implies Limit[f[x1,…,xn],{x1,…,xn}{∞,…,∞}]f* for every there is such that implies • • 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 \$Assumptions assumptions on parameters Direction Reals directions to approach the limit point GenerateConditions Automatic whether to generate conditions on parameters Method Automatic method to use PerformanceGoal "Quality" aspects of performance to optimize
• Possible settings for Direction include:
•  Reals or "TwoSided" from both real directions "FromAbove" or -1 from above or larger values "FromBelow" or +1 from below or smaller values Complexes from 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:
•  Automatic non-generic conditions only True all conditions False no conditions None return 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:

 In:= Out= In:= Out= Limit at infinity:

 In:= Out= In:= Out= Limit from above:

 In:= Out= Limit from below:

 In:= Out= The two-sided limit does not exist:

 In:= Out= In:= Out= ## Neat Examples(2)

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