Before You Begin

Question #41423

Difficulty: 8/10 | 🟢 Free

The Question

Let $\omega$ be a complex cube root of unity ( $\omega \neq 1$ ). Evaluate the following limit: $\lim_{x \to 0} \frac{ \sin^2 x + \omega^2 x \sin x + \omega x^2 }{ \left( \cos^2 x + \omega^2 \cos x + \omega \right) \left( \frac{\sin(2x)\cos(5x/2)}{\sin(x/2)} - 4 \right) }$

Options

A.
$-\frac{1}{45}$
B.
$-\frac{1}{15}$
C.
0
D.
$\frac{1}{45}$

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

Trigonometric Identical Substitutions|Lagrange Identity Equivalents|Quadratic Equations with Complex Coefficients|Trigonometric Standard Limits

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Lagrange's trigonometric identity with complex-coefficient limits.

The primary trap lies in the quadratic form $\sin^2 x + \omega^2 x \sin x + \omega x^2$ .

Students habitually apply the standard limit $\sin x \approx x$ prematurely, which yields a $0$ numerator and leads to an incorrect conclusion of $0$ or an 'undefined' $0/0$ state.

Furthermore, failing to factorize the complex quadratic prevents the isolation of the $(1-\omega)$ term, trapping students in endless L'Hopital loops.

The Lagrange identity $\sum \cos(kx)$ must be recognized in reverse to evaluate the high-degree trigonometric denominator.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Trigonometric Identical Substitutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free