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Question #40514

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the following limit: $\lim_{x \to 0^-} \frac{e^{x|x|} - \cos x + x^3 \sin\left(\frac{1}{x}\right)}{x |\sin x|}$

Options

A.
$\frac{1}{2}$
B.
$-\frac{1}{2}$
C.
$\frac{3}{2}$
D.
\text{Does not exist}

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

Sign Flipping Rule|L'Hopital's Pitfall|Domain Restrictions|Squeeze Theorem

Why This Question Is Tricky

Common Mistakes

This question tests the lethal collision between L'Hopital's Rule, domain-restricted sign flipping, and the Squeeze Theorem.

Students who blindly apply L'Hopital's Rule will encounter a non-existent limit due to the oscillating $x^3 \sin(1/x)$ term, leading them to incorrectly choose 'Does not exist'.

Furthermore, the left-hand limit forces $x < 0$ , triggering the sign flipping rule where $|x| = -x$ and $|\sin x| = -\sin x$ .

Failing to apply this domain restriction before expansion yields incorrect limits like $3/2$ or $-1/2$ .

The Squeeze Theorem acts as a secondary filter, requiring students to isolate the bounded oscillating term rather than differentiating it.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Sign Flipping Rule
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free