**Contents:**

## 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:

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

- (x
_{1}, x_{4}, x_{7}, …) converges to 1. - (x
_{2}, 5_{1}, 8_{1}, …) converges to 0. - (x
_{3}, x_{6}, x_{9}, …) converges to -1.

## Examples: Finding a Subsequential Limit

**Example question #1:** Find the subsequential limits of the sequence a_{n} = 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:

- Type the numbers 1 through 10 in column A of a worksheet.
- Type the formula in cell B2. When you get to the “n”, click on cell A1.
- Press Enter, then grab the little square at the right hand corner of B2. Drag this square to the bottom of column B2.
- The
**limit inferior**is the*smallest*limit of a subsequence. - The
**limit superior**is the*largest*limit of a subsequence.

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 a_{n} = 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:

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 |a_{n}| = 1/n → -, so a_{n} → 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]:

For bounded sequences, these limits always exist [2].

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 a_{i} = (-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 [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).

## Formal Definition

The limit inferior for a sequence x_{n}can 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.

## 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/

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!