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Question #40675

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sum_{n=1}^{\infty} x^3 (1 + x|x|)^{n-1}$ . Evaluate the limit: $L = \lim_{x \to 0} \left( 1 + f(x) \right)^{\frac{1}{|x|}}$

Options

A.
$e^{-1}$
B.
$e$
C.
$1$
D.
Does not exist

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

Infinite GP Convergence|The 1^∞ Form Shortcut|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the convergence criteria of an infinite geometric progression embedded within a limit problem.

The trap lies in the algebraic simplification of the infinite sum formula $S_\infty = a/(1-r)$ , which simplifies to $-|x|$ and perfectly masks the domain restriction.

Students who blindly apply the $1^\infty$ shortcut without verifying the condition $|r(x)| < 1$ will incorrectly calculate the limit as $e^{-1}$ .

The modulus function creates a lethal collision by ensuring the GP converges on the left side of zero but diverges on the right, rendering the two-sided limit non-existent.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Infinite GP Convergence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free