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
What is included
See a concise answer and full derivation for evaluate \(\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}\).
Skills assessed
- Rationalization
Prerequisites
- function notation
- algebraic factoring
Printable preview
- Evaluate .
Complete worked solutions
Every problem has a source-matched answer and independently reviewed derivation.
01
Problem 1: Evaluate .
Answer:
Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.
- Multiply numerator and denominator by the conjugate .
- Use the difference-of-squares identity and cancel the factor that tends to zero.
- Substitution in the simplified expression gives .
Common errors
- Multiplying by a conjugate without applying it to both numerator and denominator.