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Question #41865

Difficulty: 8/10 | 🟢 Free

The Question

Let $D$ be the set of all real numbers $x$ satisfying the inequality $|x^2 - 4| \le 5|x - 2|$ . Consider the limit $L = \lim_{x \to 2} \frac{\sin(x^2 - 4)}{|x^2 - 4| - 5|x - 2|}$ evaluated over the domain $D$ . Find the value of $L$ .

Options

A.
4
B.
-4
C.
0
D.
\text{Does not exist}

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

Modulus Sign Dependency|System of Inequalities|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle limits involving absolute values and the common pitfall of blindly applying L'Hôpital's rule.

The domain $D$ ensures $x=2$ is an interior point, requiring both left and right limits.

Students often ignore the piecewise nature of the modulus function, incorrectly assuming differentiability at the boundary $x=2$ .

If they apply L'Hôpital's rule without resolving the modulus signs based on the approach direction, they will erroneously obtain 4 or -4 (depending on which sign branch they implicitly assume), falling for the distractors.

The true limit does not exist because the left-hand and right-hand limits diverge due to the sign flip in the denominator's non-differentiable terms.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Modulus Sign Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free