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Limit Inferior, Limit Superior and Subsequential Limits

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Contents:

  1. Subsequential Limits
  2. Limit Inferior & Limit Superior

Subsequential Limits

A subsequential limit is the limit of a subsequence; a subsequence is a smaller part of some larger sequence. For example, the integer sequence {1, 2, 1, 2} has two subsequences: one for odd numbers and one for even numbers.

Sometimes a limit doesn’t exist for a sequence, but does exist for one or more subsequences. For example, the limit for the following sequence doesn’t exist:
sequence with no limit

However, there are three subsequences with limits [1]:


  • (x1, x4, x7, …) converges to 1.
  • (x2, 51, 81, …) converges to 0.
  • (x3, x6, x9, …) converges to -1.

Examples: Finding a Subsequential Limit

Example question #1: Find the subsequential limits of the sequence an = 1 – (-1)n.

The key here is to look for a pattern by generating some terms.

Step 1: Generate some terms of the sequence. One of the easiest ways to do this is in Excel. Here’s the steps:

  1. Type the numbers 1 through 10 in column A of a worksheet.
  2. Type the formula in cell B2. When you get to the “n”, click on cell A1. generating terms of a sequence in excel
  3. Press Enter, then grab the little square at the right hand corner of B2. Drag this square to the bottom of column B2.
  4. Step 2: Identify the pattern. We have two subsequences here: odd numbers where the limit is 2 and even numbers where the limit is 0. Therefore, the subsequential limits are 0 and 2.

    Example question #2: Find the subsequential limits of the sequence an = 1 – (-1)n.

    Step 1: Generate some terms of the sequence. For this example, I didn’t see a pattern for the first ten terms, so I went much larger:
    example 2 subsequence limit

    Step 2: Identify the pattern.The sequence (and all of its subsequences) converge to 1. This is an example where it may be easier to demonstrate the limit with algebra. We know that |an| = 1/n → -, so an → 1.

    The limits of subsequences can more formally be defined by the limit inferior and limit superior.

    Limit Inferior & Limit Superior

    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
    [4] Upper and Lower Limits of Sequences of Real Numbers. https://faculty.math.illinois.edu/~r-ash/RV/RV3.pdf

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