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Question #40919

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_{x}^{x^2} \sqrt{1 - \cos(2\pi t)} \, dt$ . If $L_1 = \lim_{x \to 1^+} \frac{f(x)}{(x-1)^2}$ and $L_2 = \lim_{x \to 1^-} \frac{f(x)}{(x-1)^2}$ , then the value of $\frac{L_1 - L_2}{\pi \sqrt{2}}$ is

Options

A.
3
B.
0
C.
6
D.
-3

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

Definition of Limit|Newton-Leibniz Formula with variable limits|Square Root and Modulus Logic|Domain and Range Nuance

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of the Newton-Leibniz formula, L'Hopital's rule, and the rigorous definition of the modulus function.

The 'Lethal Collision' occurs because the limit approaches $x=1$ , forcing the integration bounds to squeeze around $t=1$ , the exact point where the integrand's modulus argument changes sign.

Students who blindly write $\sqrt{1-\cos(2\pi t)} = \sqrt{2}\sin(\pi t)$ without the absolute value will incorrectly find $L_1 = L_2$ , leading to an answer of 0.

The problem demands splitting the integrand's definition based on the directional trajectory of the limit (left-hand vs right-hand), revealing a hidden jump discontinuity in the second derivative of the integral.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Definition of Limit
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free