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Use conjugates to evaluate limits involving square roots
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When you have square roots in a limit that produces , factoring won't help. Instead, use rationalizing with conjugates.
The conjugate of is (flip the sign between terms).
Evaluate
| Section | Format | Questions | Time | Weight | Calculator |
|---|---|---|---|---|---|
| Multiple Choice (No Calculator) | MCQ | 30 | 60 min | 33.3% | ๐ซ |
| Multiple Choice (Calculator) | MCQ | 15 | 45 min | 16.7% | โ |
| Free Response (Calculator) | FRQ | 2 | 30 min | 16.7% | โ |
| Free Response (No Calculator) | FRQ | 4 | 60 min | 33.3% | ๐ซ |
Avoid these 4 frequent errors
See how this math is used in the real world
A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of cm/s. How fast is the area of the circle increasing when the radius is cm?
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Key property:
This eliminates the square root!
Direct substitution: โ
Multiply by conjugate :
Simplify numerator:
Cancel :
Direct substitution:
Multiply by conjugate of denominator:
Conjugates turn differences into factored forms, eliminating radicals!
Multiply by conjugate:
Simplify numerator:
Cancel :
Direct substitution:
Answer: 1/6