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Question #41511

Difficulty: 8/10 | 🟢 Free

The Question

Let $I_A(t)$ denote the indicator function of the set $A$ , which equals $1$ if $t \in A$ and $0$ otherwise. For $x > 0$ , define the function: $P(x) = \int_{\mathbb{R}} \frac{2024^{t^2}}{2024^{t^2} + 2024^{(x-t)^2}} I_{[0, x]}(t) dt$ Evaluate the limit: $L = \lim_{x \to 0^+} \frac{\int_{0}^{x} t \cdot P(t) \cdot \sin(2t) dt}{1 - \cos(x^2)}$

Options

A.
$\frac{1}{4}$
B.
$\frac{1}{2}$
C.
1
D.
2

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

Definite Integral Properties (King's Property)|Indicator Function Logic|Equiprobability Fallacy|Indeterminate Forms

Why This Question Is Tricky

Common Mistakes

This question tests the application of King's Property disguised by an indicator function, followed by resolving an indeterminate form using the Leibniz rule.

The primary trap is the 'Equiprobability Fallacy', where students misinterpret the indicator function as implying a uniform density $P(t)=1$ or fail to divide by 2 after applying King's Property, leading to incorrect limits like $\infty$ or $1$ .

Furthermore, careless application of L'Hôpital's rule on the composite trigonometric denominator often results in missing the factor of 2 from the chain rule, trapping students into choosing option C or D.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Definite Integral Properties (King's Property)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free