Before You Begin

Question #40933

Difficulty: 8/10 | 🟢 Free

The Question

Let $L = \lim_{x \to 0} \left( \sum_{k=0}^{\lfloor 1/x^2 \rfloor} (24 x \cdot \text{sgn}(x))^k \right)^{\frac{4}{x} \sum_{n=1}^{\infty} \frac{1}{n(n+2)}}$ Find the value of $L$ , where $\lfloor \cdot \rfloor$ denotes the greatest integer function and $\text{sgn}(\cdot)$ is the signum function.

Options

A.
$e^{96}$
B.
$e^{72}$
C.
$e^{24}$
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

The 1^∞ Form Shortcut|Infinite GP Convergence|The Telescopic Series Trap|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

The question tests the rigorous application of the $1^\infty$ limit form by embedding two distinct traps.

First, the 'Telescopic Series Trap' tempts students to incorrectly evaluate the sum as $1$ (due to early cancellation of terms), leading to a false exponent and the bait answer $e^{96}$ .

Second, the base incorporates $x \cdot \text{sgn}(x) = |x|$ , creating a non-differentiable point at $x=0$ .

Shortcut-reliant students who blindly apply the exponential limit formula without separating into Left-Hand and Right-Hand limits will falsely conclude the limit is $e^{72}$ or $e^{96}$ .

The absolute value forces the LHL to be $e^{-72}$ and RHL to be $e^{72}$ , meaning the actual limit does not exist.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	The 1^∞ Form Shortcut
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free