Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f(x)$ be a differentiable function, $f(1) = 0, f'(1) = 2$ then the value of $\lim_{x \to 1}\frac{\int_1^x \sin(t(f(t)))dt}{(x-1)^2}$ is_____.
Step-by-Step Solution
Key Concept: Use L'Hôpital's rule to evaluate limits involving integrals and products by differentiating numerator and denominator.
Apply L'Hôpital's rule to the limit as $x \to 1$. The numerator involves $x^2 \sin(f(x^2)) ≈ 2x + 2\int_0^{x^2} \sin f(t)dt$ and denominator is $2(x-1)$. After applying L'Hôpital's rule, differentiate to get $2f'(1) = 4$, so $f'(1) = 2$.
Correct Answer: 2