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Limit Inferior and Limit Superior

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The limit inferior and limit superior of a sequence give bounds on the sequence’s subsequential limits (i.e. the limit of a subsequence) [1]:

  • The limit inferior is the smallest limit of a subsequence.
  • The limit superior is the largest limit of a subsequence.

For bounded sequences, these limits always exist [2].
In notation, the limit inferior may be written as:
limit inferior

The limit superior may be written as:
limit superior

Visual Example of Limit Inferior and Limit Superior

The bounded sequence ai = (-1)i(i + 1)/i is not convergent.
However, two subsequences (odd terms and even terms) are convergent, as shown on the following graph:
bounded sequence

From the graph, we can see that the subsequence of even terms converges to 1, which means that the limit superior = 1. The limit inferior converges from below to -1. These limits give a qualitative measure of a sequence’s asymptotic behavior [3].

Another example is the (relatively) famous divergent sequence (1, -1, 1, -1, 1,…). While the sequence as a whole does not converge, the even terms converge to -1 (i.e. lim inf = 1) and the odd terms converge to 1 (lim sup = 1).

graph of divergent sequence 1 -1 1 -1

Graph of divergent sequence 1, -1, 1, -1,… with limit inferior (red) and limit superior (black).


Formal Definition

The limit inferior for a sequence xncan more formally be defined as follows:
lim inf definition 2

The limit superior can be defined in a similar way:
lim sup definition


The limit inferior is always smaller than the limit superior, unless the sequence is convergent. If that happens, then the two limits are equal. In notation, we can say:

lim inf ≤ lim sup.

References

Sequence Images: Desmos.
[1] Basic properties of limit inferior and limit superior. Retrieved May 3, 2021 from: https://www.uio.no/studier/emner/matnat/math/MAT1100/h20/grublelimsup.pdf
[2] Lebl, J. Basic Analysis I & II: Introduction to Real Analysis, Volumes I & II. 2.3 Limit superior, limit inferior, and Bolzano–Weierstrass. Retrieved May 3, 2021 from: https://www.jirka.org/ra/html/sec_bw.html
[3] Chidume Chapter 5. Retrieved May 3, 2021 from: http://www.math.utoledo.edu/~dwhite1/d_makerere/Chidume2.pdf


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Stephanie Glen. "Limit Inferior and Limit Superior" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/limit-inferior-and-limit-superior/
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