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Question #40533

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the following limit: $L = \lim_{x \to 0} \frac{x^2 \sin x - x \ln(1+x^2)}{(\tan x - \sin x)^2}$

Options

A.
$\frac{4}{3}$
B.
0
C.
$\frac{3}{4}$
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

Even Multiplicity Roots|The L'Hôpital's Rule Misuse|Chain Rule Complexity

Why This Question Is Tricky

Common Mistakes

This question targets the lethal collision between L'Hôpital's rule misuse and Taylor series truncation errors.

Students encountering the $0/0$ form often blindly apply L'Hôpital's rule, but the even multiplicity of the denominator's root (due to the square) creates immense Chain Rule complexity, forcing algebraic fatigue.

Those who switch to Taylor series often fall into a conceptual trap: they extract the leading coefficients ( $1/3$ and $1/4$ ) and incorrectly conclude the limit is $4/3$ .

However, they fail to notice that the numerator vanishes at order 5 while the denominator vanishes at order 6.

This leaves a $1/x$ factor, causing the function to blow up to $+\infty$ on the right and $-\infty$ on the left, meaning the limit strictly does not exist.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Even Multiplicity Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free