AP® Calculus AB · Unit 1 · Topic 1.6
Rationalizing and Limits with Radicals
When a limit with a square root gives 0/0, multiply numerator and denominator by the conjugate radical expression. This turns the radical into a difference of squares, which cancels the troublesome factor and reveals the real limit.
Direct substitution is always your first move, but sometimes it hands you 0/0 — not an answer, just a dead end that means more algebra is needed. With square roots, that algebra is rationalizing: multiply the whole expression by a fraction equal to 1, built from the conjugate of the radical part. If the numerator is √(x+4) - 2, its conjugate is √(x+4) + 2. Multiplying top and bottom by that conjugate turns (√(x+4) - 2)(√(x+4) + 2) into (x+4) - 4, which is just x. No more square root.
Once that x appears in the numerator, it cancels with the x already sitting in the denominator. What's left is a tidy expression — no zero in the denominator anymore — so you can plug in the x-value directly and get a real number. The square root didn't vanish entirely; it just moved into a part of the expression that no longer causes trouble at that point.
This is the same rescue move as factoring: both exist because 0/0 means the original expression and its simplified form agree everywhere except at one troublesome point, and limits only care about behavior near that point, not at it. If the radical shows up in the denominator instead, the strategy doesn't change — you still multiply by the conjugate of whatever has the square root, you just watch which factor ends up canceling. This topic is the square-root cousin of the factoring techniques from earlier in the unit, and together they're the main toolkit for beating indeterminate forms before limits moves on to more automatic methods.
Key terms
- Conjugate
- The expression formed by flipping the sign between two terms, such as √(x+4) + 2 for √(x+4) - 2.
- Rationalizing the numerator
- Multiplying top and bottom by a conjugate so the square root in the numerator is eliminated.
- Indeterminate form
- An expression like 0/0 that gives no information about the limit until it is simplified further.
- Difference of squares
- The identity (a-b)(a+b) = a2 - b2, which removes square roots when multiplied by a conjugate.
Watch it explained
- How it worksHow does multiplying by the conjugate turn a stubborn radical into something that cancels?
- Worked exampleFind the limit of (√(x + 4) - 2)/x as x approaches 0 by rationalizing the numerator.
- Common mistakeWhy isn't direct substitution enough here even though the expression looks harmless?
- Going furtherWhat happens when the radical sits in the denominator instead of the numerator?
Check yourself
End of topic.