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Question #41901

Difficulty: 8/10 | 🟢 Free

The Question

Let $f_k(x) = \lim_{t \to 0} \frac{\sin(xt)}{t \sqrt{1+kx^2}}$ for $k \in \mathbb{R}$ , where the domain of $f_k$ is the maximal subset of $\mathbb{R}$ for which the limit exists and is a real number. Let $g(x) = f_a(f_b(x))$ for some constants $a, b \in \mathbb{R}$ . What is the necessary and sufficient condition on $a$ and $b$ such that the maximal domain of $g(x)$ is $\mathbb{R}$ ?

Options

A.
a + b \ge 0
B.
b \ge 0 \land a + b \ge 0
C.
a \ge 0 \land b \ge 0
D.
a \ge 0 \land a + b \ge 0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Invertibility in Binary Operations|Algebra of Functions|Trigonometric Standard Limits

Why This Question Is Tricky

Common Mistakes

This question tests the algebra of functions and domain restrictions in composition.

Students often simplify $f_a(f_b(x))$ algebraically to $\frac{x}{\sqrt{1+(a+b)x^2}}$ and incorrectly conclude that the domain is $\mathbb{R}$ as long as $a+b \ge 0$ (Option A).

This is a classic trap.

They forget that the domain of a composite function $f(g(x))$ is strictly a subset of the domain of the inner function $g(x)$ .

Therefore, $f_b(x)$ must itself have a domain of $\mathbb{R}$ , which imposes the hidden necessary condition $b \ge 0$ .

The problem elegantly combines trigonometric limits with non-trivial domain analysis.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Invertibility in Binary Operations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free