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Question #40378

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{x^2 + [x]}{x^2 + 1}$ for $x > 0$ , where $[ \cdot ]$ denotes the greatest integer function. Let $L = \lim_{x \to \infty} f(x)$ and $S = \{x > 0 \mid f'(x) < 0\}$ . Which of the following represents the ordered pair $(L, S)$ ?

Options

A.
$(1, (2, \infty) \setminus \mathbb{N})$
B.
$(1, (1, \infty) \setminus \mathbb{N})$
C.
$(0, (2, \infty) \setminus \mathbb{N})$
D.
$(1, (0, \infty) \setminus \mathbb{N})$

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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

Rational Function Asymptotes|Increasing/Decreasing Strictness|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to handle limits and derivatives of rational functions containing the Greatest Integer Function (GIF).

The primary trap is assuming that because $\lim_{x \to \infty} f(x) = 1$ , the function behaves monotonically like its continuous counterpart $\frac{x^2+x}{x^2+1}$ .

Furthermore, students often blindly differentiate without considering the piecewise constant nature of $[x]$ , leading them to incorrectly include $(1,2)$ in the strictly decreasing set, failing to realize that when $[x]=1$ , the numerator and denominator are identical, making the function locally constant ( $f'(x)=0$ ).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Rational Function Asymptotes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free