**Contents**:

In a nutshell, the **algebraic limit theorem** and **order limit theorem** are very similar in that they both describe limits for bounded convergent sequences. While the algebraic theorem uses +, –, *, and ÷, the order theorem uses ≤ and ≥.

## What is the Algebraic Limit Theorem?

The **algebraic limit theorem** shows us that sequences behave in an orderly fashion when subjected to mathematical operations.

Formally stated, the algebraic limit theorem is [1]:

If *a* = lim *a*_{n} and *b *= lim *b*_{n}, then

- lim(
*ca*_{n}) =*ca*for all*c*∈ ℝ, - lim(
*a*_{n}+*b*_{n}) =*a*+*b*, - lim(
*a*_{n}*b*_{n}) =*ab*, - lim(
*a*_{n}/*b*_{n}) =*a/b*provided b ≠ 0.

Each of these statements has their own proof. For example, the proof for the first statement is as follows:

**Proof** [2]: Assuming that *c* ≠ 0, the goal is to show the sequence converges to *c* · *a*.

- Let ε = an arbitrary positive number. We want to find the point in the sequence where |
*ca*_{n}–*ca*| < ε. - |
*ca*_{n}–*ca*| = |*c*| |*a*_{n}–*a*| - Choose an N so that:

This is true whenever n ≥ N. Here, we’re making N very, very small. - Show that N works. For all n ≥ N:

## Algebraic Limit Theorem Example: A Worked Proof [3]

**Example question:** Show that If (x_{n}) → 2, then ((2x_{n} – 1)/3) → 1.

- Start with the given information: X
_{n}→ 2. - Rewrite as a pair of fractions:

- Let:

- Substitute the values into the algebraic limit theorem, which tells us that ca
_{n}→ ca. This results (with a little numerical manipulation) in:

- Let:

- We know by the Algebraic Limit Theorem that a
_{n}+ b_{n}→ a + b., so:

## Alternative Definition for Functions

The algebraic limit theorem for functions is similar to the basic definition, except it is rewritten in function notation. For a limit x→p, the theorem is:

- lim (f(x) + g(x)) = lim f(x) + lim g(x)
- lim (f(x) – g(x)) = lim f(x) – lim g(x)
- lim (f(x) · g(x)) = lim f(x) · lim g(x)
- lim (f(x) / g(x)) = lim f(x) / lim g(x)

The indeterminate limits (on the left) may exist even when the limits on the right do not. If any of the theorems don’t give you a limit, try an alternate method like the sandwich theorem or L’Hospital’s rule.

## What is the Order Limit Theorem?

Suppose *a *= lim *a*_{n} and *b *= lim *b*_{n}. The order limit theorem states that [1]:

- If
*a*_{n}≥ 0 for all*n*∈ ℕ, then*a*≥ 0. - If
*a*_{n}≤*b*_{n}for all*n*∈ ℕ, then*a*≤*b*. - Assuming there is a
*c*∈ ℝ:- If
*c*≤*b*_{n}for all*n*∈ ℕ, then*c*≤*b*. - If
*a*_{n}≤*c*for all*n*∈ ℕ, then*a*≤*c*.

- If

## References

[1] Bakker, L. Math 341 Lecture #8. §2.3: The Algebraic and Order Limit Theorems. Retrieved August 19, 2021 from: https://math.byu.edu/~bakker/M341/Lectures/Lec08.pdf

[2] Chapter 2: Sequence and Series. Retrieved August 19, 2021 from: http://econdse.org/wp-content/uploads/2018/08/sequence-reading.pdf

[3] Lytle, B. Introduction to the convergence of sequences. Retrieved August 19, 2021 from: https://math.uchicago.edu/~may/REU2015/REUPapers/Lytle.pdf