1.5 Limits Using Algebraic Properties

Limit laws

Limits distribute over operations: the limit of a sum is the sum of limits; of a product, the product of limits; of a quotient, the quotient (as long as the denominator's limit is not 0); and a constant multiple pulls out front.

Direct substitution

For functions continuous at c (like polynomials), lim(x→c) f(x) = f(c) — just plug in. This is called direct substitution and is always the first thing to try.

Key terms

Limit laws

Limits distribute over sums, differences, products, quotients, and powers.

Sum/product rules

Limit of a sum/product = sum/product of the limits.

Constant multiple rule

lim k·f(x) = k·lim f(x).

Direct substitution

For f continuous at c, lim(x→c) f(x) = f(c).

Quotient rule condition

The denominator's limit must be nonzero.

Try an AP-style question

If lim(x→a) f(x) = 3 and lim(x→a) g(x) = −2, what is lim(x→a) [2f(x) − (g(x))²]?

1. 8
2. 10
3. −10
4. 2