AP® Calculus AB · Unit 1 · Topic 1.5
Determining Limits Using Algebraic Properties
The limit laws say you can find the limit of a sum, product, or quotient by taking the limits of the pieces first, then combining them the same way. This turns most limits into simple substitution.
A complicated function is usually built from simple pieces glued together with addition, multiplication, or division. The limit laws say the glue doesn't care about the limiting process: the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits, as long as the bottom limit isn't zero. That's why you can often just plug in the number x is approaching. For polynomials and rational functions with a nonzero denominator at that point, direct substitution works because the function is continuous there, and the limit laws are exactly what justify skipping any extra work.
The laws break down the instant a denominator limit comes out to zero. Dividing by a limit of zero makes the quotient law inapplicable, not just risky, so plugging in and writing 0/0 is not an answer. That's a signal to factor, cancel common factors, rationalize, or otherwise rewrite the expression so the problem denominator disappears before you take the limit again. If one of the pieces has no limit at all, like 1/x as x approaches 0, the laws simply don't apply to that piece, and you can't combine your way around a limit that doesn't exist; you need direct analysis of the function's behavior instead.
This topic is the toolkit that makes every later limit technique possible. Factoring and canceling to beat a 0/0 form, which comes right after this, only makes sense once you know why substitution fails and what the quotient law actually requires.
Key terms
- Limit law
- A rule stating that the limit of a combination (sum, product, quotient) of functions equals that same combination of their individual limits.
- Direct substitution
- Plugging the approached value into the function to get the limit, valid when the function is continuous there.
- Indeterminate form
- An expression like 0/0 that doesn't tell you the limit's value and signals you need another method.
- Quotient law
- The limit of a quotient equals the limit of the numerator divided by the limit of the denominator, provided the denominator's limit is nonzero.
Watch it explained
- How it worksHow do the sum, product, and quotient limit laws let you evaluate by substitution?
- Worked exampleFind the limit of (3x2 + 2x - 1)(x + 4) as x approaches 1 using the limit laws.
- Common mistakeWhy isn't the quotient law usable when the bottom limit turns out to be zero?
- Going furtherWhat happens to the limit laws when one of the pieces has no limit at all?
Check yourself
End of topic.