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Question #41857

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = a x^3 \left\lfloor \frac{1}{x^2} \right\rfloor + b x$ and $g(x) = \sin x$ , where $a, b \in \mathbb{R}$ and $\lfloor \cdot \rfloor$ denotes the greatest integer function. The true value of the limit $\lim_{x \to 0} \frac{f(x)}{g(x)}$ is $3$ . Let $h(x) = f'(x)$ for all $x$ where $f$ is differentiable. If $\lim_{x \to 0} \frac{h(x)}{g'(x)} = 7$ (taken over the domain of $h$ ), find the value of $a^2 + b^2$ .

Options

A.
5
B.
10
C.
13
D.
17

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

Algebra of Functions|Mean Value Theorem Applicability|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question targets Trap 248: L'Hôpital's Rule Misuse.

Students often blindly apply L'Hôpital's rule without verifying the conditions of the Mean Value Theorem (specifically, differentiability in a punctured neighborhood).

Here, $f(x)$ has infinitely many jump discontinuities near $x=0$ , making L'Hôpital's rule fundamentally inapplicable.

The problem explicitly contrasts the true limit (found via the Squeeze Theorem and fractional part) with the erroneous limit obtained by blindly differentiating the piecewise constant floor function, forcing the student to recognize the trap and quantify the discrepancy.

Quick Facts About This Question

Question Information

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