DiscreteLimit

DiscreteLimit[f[k],k∞]

gives the limit _k∞f(k) for the sequence f[k] as k tends to infinity over the integers.

DiscreteLimit[f[k₁,…,k_n],{k₁,…,k_n}]

gives the nested limit ⋯ f(k₁,…,k_n) over the integers.

DiscreteLimit[f[k₁,…,k_n],{k₁,…,k_n}{,…,}]

gives the multivariate limit f(k₁,…,k_n) 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 k_i 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,…,k_n}{,…,}] 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 $TemplateBox[{{{f, (, k, )}, -, {f, ^, *}}}, Abs]<epsilon$
	DiscreteLimit[f[k₁,…,k_n],{k₁,…,k_n}{∞,…,∞}]f^*	for every there is a such that implies $TemplateBox[{{{f, (, {{k, _, 1}, ,, ..., ,, {k, _, n}}, )}, -, {f, ^, *}}}, Abs]<epsilon$

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.