AP Calculus AB and BC glossary

Conjugate method

Also called: Rationalizing technique

The conjugate method resolves a limit of the form zero over zero that contains a square root. Multiply the numerator and denominator by the conjugate of the radical expression, which turns a difference of square roots into a difference of squares and lets the offending factor cancel.

\left(\sqrt{a} - \sqrt{b}\right)\left(\sqrt{a} + \sqrt{b}\right) = a - b

For $\lim_{x \to 0}\frac{\sqrt{x+1}-1}{x}$ , direct substitution gives $\frac{0}{0}$ . Multiplying above and below by $\sqrt{x+1}+1$ turns the top into $x$ , which cancels the bottom and leaves $\frac{1}{2}$ .

The mistake

Expanding the conjugate factor you multiplied by. Leave it factored in the denominator, because the whole point is that the numerator simplifies and cancels against the original denominator.

Appears in: Unit 1: Limits and Continuity