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Question #40716

Difficulty: 8/10 | 🟢 Free

The Question

Let $F(x) = \int_0^x \frac{t^2 - x^2}{\sqrt{1+t^2}} \, dt$ . For each integer $n \ge 1$ , $n$ arithmetic means are inserted between $0$ and $F'(n)$ . If $d_n$ denotes the common difference of this arithmetic progression, find the value of $\lim_{n \to \infty} \frac{d_n}{\ln(n)}$ .

Options

A.
-2
B.
0
C.
-1
D.
2

Check Your Answer

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

Insertion of Means|Fundamental Theorem of Calculus|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Common Mistakes

This question deploys the 'Zero-Factor Cancellation Trap' within the context of the Leibniz Integral Rule.

A common student error is to blindly apply the Fundamental Theorem of Calculus by merely substituting the upper limit into the integrand, which yields $F'(x) = 0$ (since $x^2 - x^2 = 0$ ).

This leads to a false common difference of $0$ and an incorrect limit.

The problem forces students to correctly differentiate under the integral sign (accounting for the parameter $x$ inside the integrand) before applying the properties of arithmetic means and evaluating a heavy-tailed limit.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Insertion of Means
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free