# DiscreteMaxLimit

DiscreteMaxLimit[f[k],k]

gives the max limit kf(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],kωk},k] for univariate f[k] and max[ω]MaxValue[{f[k1,,kn],k1ωknωki},{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

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## Basic Examples(4)

Max limit of a sequence:

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Max limit of a product:

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Use dMlim to enter the template and to move from the underscript to the body:

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