DiscreteMaxLimit
DiscreteMaxLimit[f[k],k∞]
gives the max limit k∞f(k) of the sequence f[k] as k tends to ∞ over the integers.
DiscreteMaxLimit[f[k1,…,kn],{k1,…,kn
}]
gives the nested max limit ⋯
f(k1,…,kn) over the integers.
DiscreteMaxLimit[f,{k1,…,kn}{,…,
}]
gives the multivariate max limit f(k1,…,kn) over the integers.
Details and Options



- DiscreteMaxLimit is also known as limit superior, supremum limit, limsup, upper limit and outer limit.
- DiscreteMaxLimit computes the smallest upper bound for the limit and is always defined for real-valued sequences. It is often used to give conditions of convergence and other asymptotic properties that do not rely on an actual limit to exist.
- DiscreteMaxLimit[f,k∞] can be entered as
f. A template
can be entered as
dMlim
, and
moves the cursor from the underscript to the body.
- DiscreteMaxLimit[f,{k1,…,kn}{
,…,
}] can be entered as
…
f.
- The possible limit points
are ±∞.
- The max limit is defined as a limit of the max envelope sequence max[ω]:
-
DiscreteMaxLimit[f[k],k∞] DiscreteLimit[max[ω],ω∞] DiscreteMaxLimit[f[k1,…,kn],{k1,…,kn}{∞,…,∞}] DiscreteLimit[max[ω],ω∞] - DiscreteMaxLimit[f[k],k-∞] is equivalent to DiscreteMaxLimit[f[-l],l∞] etc.
- The definition uses the max envelope max[ω]MaxValue[{f[k],k≥ω∧k∈
},k] for univariate f[k] and max[ω]MaxValue[{f[k1,…,kn],k1≥ω∧⋯∧kn≥ω∧ki∈
},{k1,…,kn}] for multivariate f[k1,…,kn]. The sequence max[ω] is monotone decreasing as ω∞, so it always has a limit, which may be ±∞.
- The illustration shows max[k] and max[Min[k1,k2]] in blue.
- DiscreteMaxLimit returns unevaluated when the max limit cannot be found.
- The following options can be given:
-
Assumptions $Assumptions assumptions on parameters GenerateConditions Automatic whether to generate conditions on parameters Method Automatic method to use PerformanceGoal "Quality" aspects of performance to optimize - 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, DiscreteMaxLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples
open allclose allBasic Examples (4)
Use dMlim
to enter the template
and
to move from the underscript to the body:
TraditionalForm typesetting:
Scope (21)
Options (6)
Applications (7)
Properties & Relations (11)
Possible Issues (1)
Neat Examples (1)
See Also
Related Guides
Introduced in 2017
(11.2)