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Question #40928

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{1}{x^3} \begin{vmatrix} \sin x & \sin 2x & \sin 3x \\ x+[x] & 3x+[2x] & 5x+[3x] \\ 1 & 3 & 5 \end{vmatrix}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ . Evaluate $\lim_{x \to 0} f(x)$ .

Options

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

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Algebraic Simplification|Determinant Scaling Property|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance against blindly applying L'Hôpital's rule or assuming continuity when the Greatest Integer Function (GIF) is embedded within a determinant.

The trap lies in the fact that for $x \to 0^+$ , the GIF terms vanish, making the second row a scalar multiple of the third row, which algebraically simplifies the determinant to exactly 0.

A student who only checks the right-hand limit or assumes $[kx] \to 0$ will incorrectly conclude the limit is 0.

However, for $x \to 0^-$ , the GIF terms evaluate to $-1$ , breaking the linear dependence and resulting in a non-zero Maclaurin expansion of $4x^3$ , yielding a left-hand limit of 4.

This forces a transition from continuous algebraic simplification to discrete LHL/RHL analysis.

Quick Facts About This Question

Question Information

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