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Question #40941

Difficulty: 8/10 | 🟢 Free

The Question

Let $I(\lambda) = \int_0^1 \frac{\lambda}{\lambda^2 + x^2} \ln\left(\frac{1}{x}\right) e^{-x} \, dx$ . Evaluate the limit $\lim_{\lambda \to 0^+} \frac{I(\lambda)}{-\ln \lambda}$

Options

A.
$\frac{\pi}{2}$
B.
$\pi$
C.
0
D.
1

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

Squeeze Theorem (Sandwich Theorem)|Improper Integrals

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of singular integrals with a parameter, specifically focusing on the trap of logarithmic divergence.

Students often incorrectly assume that the unbounded nature of $\ln(1/x)$ near zero makes the entire integral diverge or behave unpredictably.

The core trap lies in failing to separate the $-\ln \lambda$ dominant term from the convergent $\ln t$ improper integral after substitution.

By applying the Squeeze Theorem (or Dominated Convergence), one can rigorously bound the residual integral, demonstrating that the logarithmic decay effectively mitigates the singularity, yielding a finite limit.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Squeeze Theorem (Sandwich Theorem)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free