Worked problem · Calculus I · Unit 1

Limit By Rationalization: Worked Example and Solution

A complete rationalization example with method choice, derivation, verification, and a common wrong approach.

1 problem8 min estimated timeFoundational to intermediate progression

Problem

Every problem has a source-matched answer and independently reviewed derivation.

Answer: $\frac16$

Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.

Multiply numerator and denominator by the conjugate $\sqrt{x}+3$ .
Use the difference-of-squares identity and cancel the factor that tends to zero.
Substitution in the simplified expression gives $\frac16$ .

See a concise answer and full derivation for evaluate limit as x approaches 9 the square root of (x) minus 3 over x minus 9.

Multiplying by a conjugate without applying it to both numerator and denominator.