Before You Begin

Question #40535

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{1}{x-k} \int_{0}^{x} (t^2 - a[t]) dt$ , where $k \in \mathbb{Z}^+$ , $a \in \mathbb{R}$ , and $[\cdot]$ denotes the greatest integer function. If the right-hand limit $\lim_{x \to k^+} f(x) = 0$ and the left-hand limit $\lim_{x \to k^-} f(x) = L$ (where $L$ is a finite real number), find the value of $a + k + L$ .

Options

A.
6
B.
8
C.
9
D.
12

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

Denominator Restriction|Leibniz Rule for Differentiation under Integral|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the lethal collision between the Leibniz rule for differentiation under the integral sign and jump discontinuities caused by the Greatest Integer Function (GIF).

Students commonly make the mistake of blindly applying L'Hôpital's rule and the Leibniz rule at $x=k$ , assuming the derivative of the integral is simply $x^2 - a[x]$ and evaluating it directly at $k$ .

However, because $[t]$ jumps at integer values, the integral's derivative is discontinuous.

The denominator restriction $x-k$ forces a $0/0$ form, requiring domain-specific left and right derivatives.

Failing to separate the right-hand limit (where $[x]=k$ ) and left-hand limit (where $[x]=k-1$ ) prevents the student from correctly identifying the parameters $a$ and $k$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Denominator Restriction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free