Before You Begin

Question #40404

Difficulty: 8/10 | 🟢 Free

The Question

Let $S(x) = \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} \tan\left(\frac{x}{2^k}\right)$ for $x \in (0, \pi)$ . Evaluate the limit: $\lim_{x \to 0} \frac{3x S(x) - x^2}{x^4}$

Options

A.
$\frac{1}{15}$
B.
$\frac{1}{45}$
C.
$\frac{2}{15}$
D.
$\frac{1}{5}$

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

Conditional Identity Constraints|The Telescopic Series Trap|Infinite Limit Behavior|Trigonometric Standard Limits

Why This Question Is Tricky

Common Mistakes

This question engineers a lethal collision between discrete telescopic summation and continuous limit evaluation.

The primary trap lies in the infinite limit of the telescopic sum's residual term, $\frac{1}{2^n}\cot(\frac{x}{2^n})$ .

Students often erroneously assume this term vanishes as $n \to \infty$ , failing to recognize the standard trigonometric limit $\lim_{\theta \to 0} \theta \cot \theta = 1$ , which yields $\frac{1}{x}$ .

Furthermore, substituting the simplified sum into the final limit creates a $0/0$ indeterminate form.

Students who prematurely attack this with L'Hopital's rule will be bogged down in cyclical derivatives of $x \cot x$ , whereas applying the Taylor series expansion of $\cot x$ elegantly bypasses the trap, rewarding structural foresight over brute-force calculation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Conditional Identity Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free