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Question #40939

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u}(x)$ and $\vec{v}(x)$ be two vector-valued functions defined for $x \neq 0$ as: $\vec{u}(x) = |x| \sin\left(\frac{1}{x}\right) \hat{i} + |x| \cos\left(\frac{1}{x}\right) \hat{j} - \left( |x| \left(\sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right)\right) + x^2 \sin\left(\frac{1}{x^2}\right) \right) \hat{k}$ $\vec{v}(x) = |x| \cos\left(\frac{1}{x}\right) \hat{i} - |x| \sin\left(\frac{1}{x}\right) \hat{j} + |x| \left(\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right)\right) \hat{k}$ If $A(x)$ represents the area of the triangle formed by the vectors $\vec{u}(x)$ and $\vec{v}(x)$ , evaluate: $\lim_{x \to 0} \frac{A(x)}{x^2}$

Options

A.
$\frac{\sqrt{3}}{2}$
B.
$\sqrt{3}$
C.
$\frac{1}{2}$
D.
\text{Limit does not exist}

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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

Squeeze Theorem (Sandwich Theorem)|Geometric Interpretation of Cross Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bypass apparent non-differentiability and wild oscillations caused by modulus and $\sin(1/x)$ functions.

The trap is attempting to compute the limit by analyzing individual cross-product components, which leads to a mess of oscillating terms that seem divergent.

By decomposing the vector into a primary part $\vec{u}_0$ and a perturbation $\vec{w}$ , and using the geometric identity $|\vec{u} \times \vec{v}|^2 = |\vec{u}|^2|\vec{v}|^2 - (\vec{u} \cdot \vec{v})^2$ , the student finds that the primary cross product magnitude is perfectly smooth ( $\sqrt{3}x^2$ ).

The Squeeze Theorem is then rigorously applied to show the perturbation vanishes faster than $x^2$ , yielding a clean limit.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Squeeze Theorem (Sandwich Theorem)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free