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Question #40921

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [0, \infty) \to \mathbb{R}$ be defined by $f(x) = \int_0^x \frac{|\sin t| - \sin t}{2} e^{\cos t} dt$ Let $C = e - \frac{1}{e}$ . For each integer $n \ge 1$ , let $a_n$ and $b_n$ be the unique positive real numbers satisfying $f(a_n) = 5C - \frac{1}{n} \quad \text{and} \quad f(b_n) = 5C + \frac{1}{n}$ Find the value of $\lim_{n \to \infty} (3a_n - b_n)$ .

Options

A.
$19\pi$
B.
$20\pi$
C.
$21\pi$
D.
$22\pi$

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|Increasing/Decreasing Strictness|Monotonicity and Boundedness|Region Ambiguity

Why This Question Is Tricky

Common Mistakes

This question targets the 'Plateau Trap' in the context of limits and monotonicity.

Students often heuristically assume that a function defined by an integral with a non-negative integrand is strictly increasing globally.

However, the region ambiguity introduced by the absolute value creates intervals where the derivative is zero, resulting in plateaus.

When evaluating the limit of the inverse sequence, students who ignore the plateau will incorrectly assume both $a_n$ and $b_n$ converge to the same point (yielding $20\pi$ or $22\pi$ ), or guess the midpoint ( $21\pi$ ).

The rigorous approach requires identifying that $a_n$ approaches the left endpoint of the plateau from below, while $b_n$ approaches the right endpoint from above, leading to the correct asymmetric limit.

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