Algebra of Limits
Mental model
The limit of a sum is the sum of the limits, as long as both limits exist.
You can split limits into smaller, easier pieces using basic algebra rules, just like distributing multiplication over addition.
Explanation
The algebra of limits provides tools to simplify complex expressions. But these rules have conditions. You can only split a limit if the individual limits exist as real numbers. For quotients, you can only split them if the denominator's limit is non-zero.
Derivation
Evaluate the limit of (x^2 + 5x) as x approaches 2
- \lim_{x \to 2} (x^2 + 5x) — limit-sum-rule — Start with the original limit.
- \lim_{x \to 2} (x^2) + \lim_{x \to 2} (5x) — limit-sum-rule — The limit of a sum is the sum of the limits.
- \lim_{x \to 2} (x^2) + 5 \lim_{x \to 2} (x) — limit-scalar-rule — Pull the constant 5 out of the second limit.
- (2)^2 + 5(2) — substitute-limit — Substitute x = 2 into the pieces.
- 4 + 10 — substitute-limit — Square the 2 and multiply 5 by 2.
- 14 — substitute-limit — Add to get the final answer.
Common mistakes
- Quotient rule division by zero: Students often split a limit into a fraction when the bottom limit is zero. You cannot split a limit into a fraction if the bottom limit is zero. Simplify the expression first.
Recall questions
- Explain the limit sum rule in your own words, as if to a classmate.
- When are you NOT allowed to use the quotient rule for limits?
- What must you do with a constant number pulled out of a limit?
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