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Question #41874

Difficulty: 8/10 | 🟢 Free

The Question

Let $\{y\}$ denote the fractional part of $y$ . Evaluate the following limit: $\lim_{x \to 3} \frac{\sum_{k=1}^{4} \left( \{ (x + \sqrt{x^2-1})^k \} + (x - \sqrt{x^2-1})^k \right) - 4}{x - 3}$

Options

A.
1868
B.
0
C.
236
D.
\text{Does not exist}

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Integral Part and Fractional Part|Sigma Notation Properties|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question tests the precise understanding of fractional parts and limits, specifically setting a trap for the misuse of L'Hôpital's rule.

A common mistake is assuming the identity $\{ (x + \sqrt{x^2-1})^k \} = 1 - (x - \sqrt{x^2-1})^k$ , which is only valid when $x$ is an integer, holds for all real $x$ .

If a student blindly applies this discrete identity to the continuous limit, the numerator identically becomes zero, leading to the incorrect answer of 0.

The correct approach requires recognizing that the greatest integer function is constant in a neighborhood of $x=3$ because the term inside is not an integer, allowing standard differentiation of the polynomial sum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Integral Part and Fractional Part
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free