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Question #41911

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x |t| \left( 8 + 3 \sin\left(\frac{1}{t}\right) \right) dt$ for $x \neq 0$ , and $f(0) = 0$ . Evaluate the limit: $L = \lim_{x \to 0} \frac{f(x)}{x|x|}$

Options

A.
4
B.
8
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

The L'Hôpital's Rule Misuse|Mean Value Theorem Applicability|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the precise applicability conditions of L'Hôpital's Rule and the handling of oscillatory discontinuities.

A common trap is blindly applying L'Hôpital's Rule to a 0/0 form; here, differentiating the numerator and denominator yields an expression containing $\sin(1/x)$ , which oscillates as $x \to 0$ .

Students will incorrectly conclude that the limit 'Does not exist' (Option D).

However, L'Hôpital's Rule only states that IF the limit of the derivatives exists, it equals the original limit.

Its failure does not preclude the original limit from existing.

By manually bounding the integral using integration by parts, the true limit is rigorously found.

The modulus sign acts as an additional layer, ensuring students cannot simply drop absolute values without checking lateral limits.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	The L'Hôpital's Rule Misuse
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free