Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
$\text{Lim}_{x \to 0^+} \left[3f\left(\frac{x^3-\sin^3 x}{x^4}\right)-f\left(\left[\frac{\sin x^3}{x}\right]\right)\right]$ where $[\cdot]$ denote greatest integer function.
Step-by-Step Solution
Key Concept: Use function properties (monotonicity) and trigonometric limits to evaluate composite function limits, noting left and right continuity.
For $x \to 0^-$: $f\left(\frac{x^3 - \sin^3 x}{x^4}\right) = f\left(\frac{x - \sin x}{x^3} \cdot (x^2 + \sin x \sin x + \sin^2 x)\right) = f(0^-) = 3$ (decreasing function). For $x \to 0^+$: $f\left(\frac{\sin^3 x}{x}\right) = f(0^+) = f(0) = 4$. The limit of $(9-4) = 5$ as $x \to 0$.
Correct Answer: 2