Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
In a $\triangle ABC$, angles $A, B, C$ are in A.P. If $f(C) = \lim_{A \to C} \frac{\sqrt{3} - 4\sin A \sin C}{A - C}$, then $f\left(\frac{\pi}{12}\right)$ is equal to______.
Step-by-Step Solution
Key Concept: Combining AP condition with sine/cosine identities and L'Hôpital's rule to evaluate limits involving trigonometric expressions.
Given $A, B, C$ are in AP, we have $2B = A + C$ and $A + B + C = 180°$, so $B = 60°$. Using $\cos B = \frac{a^2 + c^2 - b^2}{2ac}$ and the sine rule identity $|\sin A - \sin C| = \sqrt{\sin^2 B - \sin A \sin C}$, we derive $2\sin\left|\frac{A-C}{2}\right| = \sqrt{3} - 4\sin A\sin C$. Applying L'Hôpital's rule as $A \to C$ gives $\lim_{A \to C} \frac{\sqrt{3} - 4\sin A\sin C}{|A - C|} = \lim_{A \to C} \frac{2\sin\left(\frac{A-C}{2}\right)}{|A-C|} = 1$, which implies $f'(x) = 0$.
Correct Answer: 4