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Question #40971

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a differentiable function on $(0, \infty)$ satisfying the differential equation $x \frac{dy}{dx} - \lambda x y = 1 - \cos x$ where $\lambda > 0$ is a constant parameter. Given that $\lim_{x \to \infty} f(x) = 0$ , the value of $\lim_{x \to 0^+} f(x)$ is:

Options

A.
$\ln\left(\frac{\lambda}{\sqrt{\lambda^2 + 1}}\right)$
B.
$\ln\left(\frac{\sqrt{\lambda^2 + 1}}{\lambda}\right)$
C.
$\frac{1}{2} \ln(1 + \lambda^2)$
D.
$-\frac{1}{2} \ln(1 + \lambda^2)$

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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

Infinite Limit Behavior|Exact Differential Recognition|Arbitrary Constants vs Parameters

Why This Question Is Tricky

Common Mistakes

This question tests the 'Arbitrary Constants vs Parameters' trap within the context of infinite limit behavior.

Students typically solve the exact differential but treat the integration constant C as a free variable, often incorrectly assuming C=0 or failing to apply the boundary condition at infinity.

The trap forces students to realize that for the solution to converge to zero as x approaches infinity, the constant C must perfectly cancel the divergent behavior of the integral, making C a specific function of the parameter lambda.

This requires evaluating an improper integral using Laplace transforms or Leibniz's rule, bridging differential equations with advanced integral calculus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Infinite Limit Behavior
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free