1.6 Limits Using Algebraic Manipulation

The 0/0 form

When direct substitution gives 0/0, that is an indeterminate form — it does NOT mean the limit fails. It means you must simplify first, then substitute.

Techniques

Common moves: factor and cancel the common factor, rationalize by multiplying by the conjugate (for square roots), or combine fractions with a common denominator. For example, lim(x→2) (x²−4)/(x−2) = lim (x+2) = 4.

Key terms

Indeterminate form 0/0

Direct substitution gives 0/0; the limit still may exist.

Factor and cancel

Cancel a common factor causing the 0/0, then substitute.

Rationalizing

Multiply by the conjugate to clear a radical.

Simplify then substitute

The standard approach for 0/0 limits.

Example

lim(x→2) (x²−4)/(x−2) = 4.

Try an AP-style question

lim(x→3) (x² − 9)/(x − 3) = ?

1. 0
2. Does not exist
3. 6
4. 3