- 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 .
In notation, the limit inferior may be written as:
The limit superior may be written as:
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:
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 .
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).
The limit inferior for a sequence xncan more formally be defined as follows:
The limit superior can be defined in a similar way:
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.
Sequence Images: Desmos.
 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
 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
 Chidume Chapter 5. Retrieved May 3, 2021 from: http://www.math.utoledo.edu/~dwhite1/d_makerere/Chidume2.pdf
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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