Answer: -1/3.
- A -1/3
- B 1/3
- C 0
- D 1/6
Correct answer: A. -1/3
Explanation: Write as (sin²x − x²)/(x² sin²x). Using sinx ≈ x − x³/6: sin²x ≈ x² − x⁴/3, so sin²x − x² ≈ −x⁴/3. Denominator ≈ x²·x² = x⁴. Limit = (−x⁴/3)/x⁴ = −1/3. The negative sign is essential - a very common sign error on this problem.
As point Q slides along the curve toward P, the secant line PQ rotates into the tangent line at P, whose slope is the derivative.
Concept context
Rate of change, differentiation rules, and applications