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Question #41193

Difficulty: 8/10 | 🟢 Free

The Question

Let $S(x) = \lim_{n \to \infty} \sum_{k=1}^n \tan^{-1}\left( \frac{x}{1+k(k+1)x^2} \right)$ . Consider the function $f(x) = \int_0^x t^3 \left( \frac{2}{\pi} S(t) + 153 t^4 \sin\left(\frac{1}{t}\right) \right) dt$ for $x \neq 0$ , and $f(0)=0$ . Let $c$ be the unique point of inflection of $f(x)$ in the interval $\left(-\frac{1}{6}, \frac{1}{6}\right)$ . Find the value of $\lim_{x \to c} \frac{f''(x) + 153 x^5 \cos\left(\frac{1}{x}\right)}{x |x|}$

Options

A.
3
B.
$\frac{3\pi}{2}$
C.
0
D.
6

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

The Sum-Difference Formula Constraints|Point of Inflection Conditions|Squeeze Theorem (Sandwich Theorem)

Why This Question Is Tricky

Common Mistakes

This problem tests the rigorous application of the Squeeze Theorem and the identification of inflection points in functions with jump discontinuities masked by integration.

Students often fail to recognize the signum function emerging from the infinite sum of inverse tangents, leading to incorrect parity and derivative evaluations.

Furthermore, standard differentiation rules fail to prove $f''(0)=0$ due to the wildly oscillating term, strictly requiring the limit definition of the derivative and the Squeeze Theorem.

The '153' distractor term is designed to cancel out in the final limit, rewarding students who trust the algebraic structure and perform careful asymptotic analysis near the origin.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	The Sum-Difference Formula Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free