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Question #40970

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [0, 5] \to \mathbb{R}$ be defined as $f(x) = \lim_{n \to \infty} \frac{\left(\frac{x}{3}\right)^{2n} \sin(\pi x) + \sin(\pi x) [x]}{\left(\frac{x}{3}\right)^{2n} + 1}$ where $[x]$ denotes the greatest integer function. The function $f$ satisfies $f(0) = f(5) = 0$ and is continuous on $[0, 5]$ , but Rolle's Theorem is not applicable due to non-differentiability at a set of points $S \subset (0, 5)$ . The sum of the elements of $S$ is:

Options

A.
6
B.
10
C.
3
D.
5

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

Infinite Limit Behavior|Rolle's Theorem Anchors|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of infinite limit behavior and Rolle's Theorem conditions, specifically targeting the differentiability constraint.

The trap is twofold: First, students often assume that the presence of the step function $[x]$ automatically implies non-differentiability at all integers (leading to the incorrect inclusion of $x=4$ ).

However, the infinite limit acts as a global constraint, completely suppressing the step function for $x > 3$ .

Second, students might incorrectly assume the transition point $x=3$ is smooth because the function is continuous and evaluates to $0$ , but the left and right derivatives differ due to the residual coefficient from the step function on the left side.

This forces a rigorous piecewise derivative check rather than relying on superficial patterns.

Quick Facts About This Question

Question Information

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