# DiscreteLimit

DiscreteLimit[f[k],k]

gives the limit kf(k) for the sequence f[k] as k tends to infinity over the integers.

DiscreteLimit[f[k1,,kn],{k1,,kn}]

gives the nested limit f(k1,,kn) over the integers.

DiscreteLimit[f[k1,,kn],{k1,,kn}{,,}]

gives the multivariate limit f(k1,,kn) over the integers.

# Details and Options

• DiscreteLimit is also known as discrete limit or limit over the integers.
• DiscreteLimit computes the limiting value of a sequence f as its variables k or ki get arbitrarily large.
• DiscreteLimit[f,k] can be entered as f. A template can be entered as dlim, and moves the cursor from the underscript to the body.
• DiscreteLimit[f,{k1,,kn}{,,}] can be entered as f.
• The possible limit points are ±.
• For a finite limit value f*:
•  DiscreteLimit[f[k],k∞]f* for every there is a such that implies DiscreteLimit[f[k1,…,kn],{k1,…,kn}{∞,…,∞}]f* for every there is a such that implies
• DiscreteLimit[f[k],k-] is equivalent to DiscreteLimit[f[-l],l] etc.
• DiscreteLimit returns Indeterminate when it can prove that the limit does not exist, and returns unevaluated when no limit can 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, DiscreteLimit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

# Examples

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## Basic Examples(4)

Limit of a sequence:

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Plot the sequence and its limit:

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Limit of a multivariate sequence:

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Plot the sequence and its limit:

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

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