Limit

analysis

Definition

Sequence Definition

Definition

Limit (Sequence)

A real number is called limit of sequence if almost all sequence elements lie within every epsilon neighbourhood of :

A number is thus the limit of a sequence if the following holds: Given any distance , an index can be found such that from this index onward, all sequence elements have a from/to that is smaller than . Such an must exist for every , no matter how small is chosen.

If there exists an epsilon neighbourhood of that does not contain infinitely many sequence elements, then cannot be the limit, then cannot be the limit of that sequence.

Every limit is a limit point of that sequence.

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Function Definition

Definition

Limit (Function)

Let be a function. The limit of as approaches is equal to if, for every , there exists a such that for all :

This is denoted as .

Sequential Definition

Alternatively, the limit can be defined using sequences (Heine’s definition):

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Uniqueness

Different Limit Points

Let and be two different limit points of sequence , then for it holds that . Hence, it is impossible for almost all to lie in both and .

Uniqueness of Limits

From this consideration it follows the uniqueness of the limit: since almost all terms of the sequence lie in every neighbourhood of the limit, a convergent sequence can have only one limit point. In the case of convergence, it therefore holds that:

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Properties

Additivity

The limit is additive, i.e.:

Multiplicativity

The limit is multiplicative, i.e.:

Homogeneity

The limit is homogeneous, i.e.:

Quotient

Examples

Find Limit of

We want to prove that , which, by definition, means that for every , there exists an such that for all :

Since is always non-negative, we can simplify the above to:

To solve the inequality for , we take the reciprocal (nothing that all quantities are positive) and get:

Taking the square root on both sides (again, all values are positive) gives:

Now, choose:

with . we can establish that for an arbitrary , there exists an such that , which is the definition of a limit.

Find Limit of “Jumping” Sequence

Let be a sequence with:

We assume that . Let be given.