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Question #40444

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the following limit: $\lim_{x \to \frac{\pi}{4}} \frac{\int_{\pi/4}^x \cos(2t) |\ln(\tan t)| dt}{\left(x - \frac{\pi}{4}\right)^3}$

Options

A.
$-\frac{4}{3}$
B.
$\frac{4}{3}$
C.
\text{Does not exist}
D.
0

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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

Monotonicity and Boundedness|Logarithmic Integration via Trig Manipulation|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the careful application of limits and L'Hôpital's rule on functions defined by integrals with absolute values.

The trap lies in the non-differentiability of the integrand at $x = \pi/4$ due to the absolute value of a logarithm that changes sign.

Students commonly ignore the absolute value or blindly apply L'Hôpital's rule, which implicitly assumes local differentiability and uniform behavior on both sides of the limit.

By doing so, they incorrectly obtain $-4/3$ .

However, evaluating the left and right limits separately reveals that the sign of the integrand flips, yielding $-4/3$ from the right and $4/3$ from the left, meaning the two-sided limit does not exist.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Monotonicity and Boundedness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free