Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f$ be a function defined on $(-1, 1)$ by $f(x) = \frac{\cos^{-1}(1-|x|^2)\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^3}$, $x \neq 0$ $(\cdot)$ is the fractional part function. Which of the following statements is correct?
1. limx→0+ f(x) exists and equals π/√2
2. limx→0- f(x) exists and equals π/4
3. f is continuous
4. limx→0- f(x) exists and equals limx→0+ f(x)
Step-by-Step Solution
Key Concept: Transform inverse trigonometric expressions using substitution and standard limits like $\lim_{x\to 0}\frac{\sin^{-1}(x)}{x} = 1$.
For $L.H.L.$ at $x=0^-$, substitute $h=-1-h$ to transform the limit. After algebraic manipulation, $\cos^{-1}(h(2-h))$ approaches $\cos^{-1}(0) = \pi/2$ and $\sin^{-1}(h)/h \to 1$, yielding $L.H.L. = \pi/4$. For $R.H.L.$, let $\cos^{-1}(1-h^2) = t$ so $h^2 = 2\sin^2(t/2)$. The limit becomes $\frac{\pi}{2}\lim_{t\to 0}\frac{t/2}{\sin(t/2)} = \pi/2$.
Correct Answer: 1,2