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Question #41916

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = x + x^4 \sin\left(\frac{1}{x}\right)$ for $x \neq 0$ and $f(0) = 0$ . Evaluate the limit: $\lim_{x \to 0} \frac{f(\tan x) - f(\sin x)}{x^3}$

Options

A.
$\frac{1}{2}$
B.
1
C.
0
D.
\text{Limit 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

Mean Value Theorem Applicability|The L'Hôpital's Rule Misuse|Trigonometric Standard Limits

Why This Question Is Tricky

Common Mistakes

This question tests the limits of L'Hôpital's Rule (LHR) and the strategic use of the Mean Value Theorem (MVT).

Students often blindly apply LHR to $0/0$ forms.

However, LHR is only valid if the limit of the derivative exists.

Here, differentiating twice leads to a $\sin(1/x)$ term that oscillates wildly, making LHR fail.

The trap is avoided by applying MVT to decouple the pathological function $f(x)$ from the trigonometric arguments, reducing the problem to a standard trigonometric limit multiplied by the continuous first derivative of $f(x)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Mean Value Theorem Applicability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free