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Question #40927

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the solution to the initial value problem $(x^2 - 1) \frac{dy}{dx} + 2x y = 3x^2 + 1, \quad f(2) = 4$ defined on its maximal interval of existence. Find the value of $\lim_{x \to -1} f(x)$ .

Options

A.
-2
B.
2
C.
0
D.
\text{Does not exist}

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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

Indeterminate Forms|Exact Differential Recognition|Arbitrary Constants vs Parameters|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the domain of solutions to initial value problems (IVPs), specifically the 'Denominator Restriction' trap.

Students will correctly identify the exact differential, solve the equation, and find the algebraic expression $f(x) = (x^2 - x + 2)/(x - 1)$ after canceling the $(x+1)$ factor.

They will then incorrectly evaluate the limit as $x \to -1$ by direct substitution (resolving the 0/0 indeterminate form), yielding -2.

The lethal trap lies in ignoring the existence and uniqueness theorem for linear ODEs: the maximal interval of existence for the IVP containing the initial point $x_0 = 2$ is strictly $(1, \infty)$ due to the singularity at $x = 1$ .

Because the solution cannot be extended across this vertical asymptote, $f(x)$ is completely undefined at and around $x = -1$ , making the limit non-existent.

Quick Facts About This Question

Question Information

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