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Question #40438

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \exp\left( \sqrt{2} \int_{0}^{x} \frac{dt}{\sin t - \cos t} \right)$ . Evaluate the limit: $L = \lim_{x \to \frac{\pi}{4}} \frac{f(x)}{\left| x - \frac{\pi}{4} \right| + \left(x - \frac{\pi}{4}\right)^2 \sin\left(\frac{1}{x - \frac{\pi}{4}}\right)}$

Options

A.
$\frac{\sqrt{2}+1}{2}$
B.
$\frac{\sqrt{2}-1}{2}$
C.
0
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

Auxiliary Angle Method|Logarithmic Integration Modulus|L'Hopital's Pitfall

Why This Question Is Tricky

Common Mistakes

This question tests the 'Logarithmic Integration Modulus' and the 'L'Hopital's Pitfall'.

The integral of the reciprocal of a trigonometric sum requires the Auxiliary Angle Method, leading to a cosecant function whose antiderivative contains a crucial absolute value.

Students often forget this modulus, which would incorrectly cause the left and right-hand limits to differ, leading them to choose 'Does not exist'.

Furthermore, the denominator includes an oscillating term $(x-\pi/4)^2 \sin(1/(x-\pi/4))$ whose derivative does not converge.

This intentionally breaks L'Hopital's rule, trapping students who blindly differentiate 0/0 forms without verifying if the limit of the derivatives exists.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Auxiliary Angle Method
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free