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Algebraic Limit Theorem: Definition, Examples

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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 an and b = lim bn, then

  1. lim(can) = ca for all c ∈ ℝ,
  2. lim(an + bn) = a + b,
  3. lim(anbn) = ab,
  4. lim(an/bn) = 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.

  1. Let ε = an arbitrary positive number. We want to find the point in the sequence where |canca| < ε.
  2. |canca| = |c| |ana|
  3. Choose an N so that:
    algebraic limit theorem

    This is true whenever n ≥ N. Here, we’re making N very, very small.
  4. Show that N works. For all n ≥ N:
    algebraic limit theorem proof

Algebraic Limit Theorem Example: A Worked Proof [3]

Example question: Show that If (xn) → 2, then ((2xn – 1)/3) → 1.

  1. Start with the given information: Xn → 2.
  2. Rewrite as a pair of fractions:
    proof example
  3. Let:
    algebraic limit theorem proof example
  4. Substitute the values into the algebraic limit theorem, which tells us that can → ca. This results (with a little numerical manipulation) in:

  5. Let:
    almost there 2

  6. We know by the Algebraic Limit Theorem that an + bn → 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:

  1. lim (f(x) + g(x)) = lim f(x) + lim g(x)
  2. lim (f(x) – g(x)) = lim f(x) – lim g(x)
  3. lim (f(x) · g(x)) = lim f(x) · lim g(x)
  4. 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.


[1] Bakker, L. Math 341 Lecture #8. §2.3: The Algebraic and Order Limit Theorems. Retrieved August 19, 2021 from:
[2] Chapter 2: Sequence and Series. Retrieved August 19, 2021 from:
[3] Lytle, B. Introduction to the convergence of sequences. Retrieved August 19, 2021 from:

Stephanie Glen. "Algebraic Limit Theorem: Definition, Examples" From Calculus for the rest of us!

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