Below you will find example sentences with "limit superior". The examples show how this phrase is used in natural context and which words often surround it.
Limit Superior in a sentence
Corpus data
- Displayed example sentences: 20
- Discovered as a combination around: limit
- Corpus frequency in the collocation scan: 17
- Phrase length: 2 words
- Average sentence length: 26.1 words
Sentence profile
- Phrase position: 7 start, 12 middle, 1 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "limit superior" has 2 words and usually appears in the middle in these examples. The average sentence has 26.1 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as and the limit superior of a, has a limit superior and a, inferior, sequence and defined stand out.
- In the phrase index, this combination connects with speed limit, superior court, limit inferior, limit inferior and exposure limit, linking the page to nearby combinations.
Example types with limit superior
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
An illustration of limit superior and limit inferior. (8 words)
In that case every set has a limit superior and a limit inferior. (13 words)
Assume that the limit superior and limit inferior are real numbers (so, not infinite). (14 words)
As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and −∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). (43 words)
For every point y between x and c, we have : As y approaches c, both and become zero, and therefore : The limit superior and limit inferior are necessary since the existence of the limit of has not yet been established. (40 words)
Schechter, Sections 7.43–7.47 For a net we put : Limit superior of a net of real numbers has many properties analogous to the case of sequences, e.g. : where equality holds whenever one of the nets is convergent. (40 words)
Example sentences (20)
Limit inferior is also called infimum limit, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.
The filter base ("of tails") generated by this sequence is defined by : Therefore, the limit inferior and limit superior of the sequence are equal to the limit superior and limit inferior of respectively.
Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does not exist.
Limit superior Limit superior and limit inferior of a net of real numbers can be defined in a similar manner as for sequences.
As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and −∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities).
For every point y between x and c, we have : As y approaches c, both and become zero, and therefore : The limit superior and limit inferior are necessary since the existence of the limit of has not yet been established.
Also note that the limit inferior and the limit superior of a set do not have to be elements of the set.
An illustration of limit superior and limit inferior.
Assume that the limit superior and limit inferior are real numbers (so, not infinite).
Definition for sequences The limit inferior of a sequence (x n ) is defined by : or : Similarly, the limit superior of (x n ) is defined by : or : Alternatively, the notations and are sometimes used.
If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as real numbers or ±∞ (i.
In that case every set has a limit superior and a limit inferior.
That is, : Similarly, the limit superior of a set X is the supremum of all of the limit points of the set.
The limit superior and limit inferior of a sequence are a special case of those of a function (see below).
Therefore, these definitions give the limit inferior and limit superior of any net (and thus any sequence) as well.
For the particular case of a metric space, this can be expressed as : where lim sup is the limit superior (of the function at point ).
However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e −n may actually be less than all elements of the sequence.
In other words, any number larger than the limit superior is an eventual upper bound for the sequence.
Schechter, Sections 7.43–7.47 For a net we put : Limit superior of a net of real numbers has many properties analogous to the case of sequences, e.g. : where equality holds whenever one of the nets is convergent.
The limit superior of the filter base B is defined as : when that supremum exists.