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Question #40380

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \lim_{t \to x} \frac{|t^2 - 9| - |x^2 - 9|}{t - x}$ . Let $S$ be the set of all real values of $x$ satisfying the inequality: $\frac{x^2 - 2x - 15}{f(x) - x} \ge 0$ Which of the following represents the set $S$ ?

Options

A.
$(0, 3) \cup [5, \infty)$
B.
$[-3, 0) \cup [5, \infty)$
C.
$(-\infty, -3] \cup (0, 5]$
D.
$(0, 3] \cup [5, \infty)$

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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

Modulus Function Symmetry|Wavy Curve (Method of Intervals)|L'Hopital's Pitfall

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly evaluate limits involving modulus functions and avoid the 'L'Hopital's Pitfall'.

A common mechanical error is to blindly apply L'Hopital's rule or differentiation rules to $|x^2 - 9|$ without considering the signum function, leading to the incorrect assumption that $f(x) = 2x$ everywhere.

This trap yields the incorrect interval $[-3, 0) \cup [5, \infty)$ .

Furthermore, the problem requires rigorous application of the Wavy Curve method across piecewise domains, ensuring students check boundary conditions where the limit (derivative) fails to exist at $x = \pm 3$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Modulus Function Symmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free