Limit Laws
Limits of sums, products and quotients follow simple rules.
About this topic
Limit laws allow limits to be computed algebraically: the limit of a sum, difference, product or quotient (denominator limit nonzero) is the sum, difference, product or quotient of the limits, and constants pull out. For polynomials, you can simply substitute.
When substitution gives 0/0, the expression is an indeterminate form and more work is needed: factor and cancel, multiply by a conjugate to rationalise, or simplify a complex fraction. For lim x→3 (x² − 9)/(x − 3), factoring gives (x − 3)(x + 3)/(x − 3) = x + 3, so the limit is 6. For lim x→0 (√(x + 4) − 2)/x, multiplying top and bottom by the conjugate gives 1/(√(x + 4) + 2), so the limit is 1/4.
Other useful facts include the squeeze theorem, which traps a function between two others sharing a limit and proves that lim x→0 sin x / x = 1.
Tips: try direct substitution first, never cancel terms that are not factors, and note that a nonzero number divided by 0 signals an asymptote, not 0/0. Limit laws let you handle complicated expressions by breaking them into simple pieces.
Key ideas
- Limit of a sum is the sum of limits
- Constants pull out
- Direct substitution works for continuous functions
Quick check
Use cases: where Limit Laws helps
- Homework: review the key idea behind Limit Laws before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Limit Laws without notes.
- Visual learning: play with the interactive lab to see how Limit Laws changes when you adjust its inputs.
- Teaching a friend: explain Limit Laws in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Limit Laws?
Limits of sums, products and quotients follow simple rules.
What will I learn in Limit Laws?
Limit of a sum is the sum of limits. Constants pull out. Direct substitution works for continuous functions.
Is Limit Laws part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Limit Laws lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Limit Laws?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.