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Question #41881

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} \sin^2 x & x^2 & x \\ \tan x & e^x - 1 & \ln(1+x) \\ \left[ x^2 \right] & \left[ 3\cos x \right] & \left[ 4\cos x \right] \end{vmatrix}$ where $[t]$ denotes the greatest integer less than or equal to $t$ . Find the value of $\lim_{x \to 0} \frac{f(x)}{x^2}$

Options

A.
1
B.
2
C.
3
D.
4

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Trigonometric Standard Limits|Differentiation of Determinants|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of determinant evaluation, standard limits, and the discontinuous nature of the greatest integer function.

The 'Lethal Collision' occurs if a student mechanically applies L'Hôpital's rule or differentiates the determinant directly at $x=0$ .

Because $[3\cos x]$ and $[4\cos x]$ have jump discontinuities at $x=0$ (taking values 3 and 4 respectively, instead of their limiting values 2 and 3), the determinant is non-differentiable.

Evaluating the entries at $x=0$ before taking the limit implicitly assumes continuity, leading to an incorrect derivative and a trap answer of 3.

The correct approach requires evaluating the greatest integer functions in a punctured neighborhood of zero first.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Trigonometric Standard Limits
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free